Powered by Blogger.

TOR signaling and cancer

Sunday, April 27, 2008

Another recent development pertinent to the discussion of TOR signaling and cancer (see here), is the announcement of preclinical findings about a potential anti-cancer drug that may act against ovarian cancer. The drug works by inhibiting the mTOR signaling pathway. (mTOR is the mammalian form of TOR.)

This is not at all the first anti-cancer drug that's come along with a similar mechanism of action. But it's still interesting, because any drug that affects TOR signaling has the potential of also causing unwanted side effects, since TOR signaling is involved in so many cell processes. Presumably some effort has been made to find reasons why the effect of the drug should be limited to cancer cells.

The drug is called NV-128, and has been developed by an Australian biotech company called Novogen. Since the drug hasn't yet entered clinical trials in humans, it could take a decade or so (as usual) to perform enough testing to determine that NV-128 is actually effective, and relatively safe.

Anyhow, here's the news release:

Drug Compound Leads To Death Of Ovarian Cancer Cells Resistant To Chemotherapy (4/17/08)
In a discovery that may be useful for maintaining remission in chemo-resistant ovarian cancer, Yale scientists report that pre-clinical studies have shown the drug compound NV-128 can induce the death of ovarian cancer cells by halting the activation of a protein pathway called mTOR.

Many traditional cancer drugs work by triggering cell death via apoptosis. Unfortunately, apoptosis needs enzymes called caspases to work, as explained here. And cancer cells may develop a circumvention of this mechanism by turning down the production of caspases, which are needed to allow mitochondria to respond to apoptosis signals. NV-128, however, is able to overcome this problem by triggering caspase-independent cell death.
In cancer cells, mTOR signals enhance tumor growth and may be associated with resistance to conventional therapies. Inhibition of mTOR could shut down many of these survival pathways, including proteins that protect the mitochondria of cancer cells.

Here's the Novogen press release:

Novogen’s NV-128 shown to target the akt-mTOR receptor in chemoresistant cancer cells (4/15/08)
NV-128 is unique in that it does not induce caspase-mediated apoptosis which can be non-functional in chemoresistant cancer cells due to accumulated mutations in tumour suppressor/promoter genes and over-expression of anti-apoptotic proteins. Rather, NV-128 uncouples the akt-mTOR­P70S6K signal transduction cascade which has a key role in driving protein translation and uncontrolled cancer cell proliferation. Further, NV-128 induces mitochondrial depolarization via a novel pathway involving the autophagy protein Beclin-1 and Bcl-2, thereby resulting in endonuclease G translocation to the nucleus and cell death.

The same research group that presented the findings just mentioned has also done work on ovarian cancer itself, and been able to locate cancer stem cells for this type of cancer:

Ovarian Cancer Stem Cells Identified, Characterized (4/17/08)
Researchers at Yale School of Medicine have identified, characterized and cloned ovarian cancer stem cells and have shown that these stem cells may be the source of ovarian cancer's recurrence and its resistance to chemotherapy.

As already mentioned, NV-128 is not the only drug under investigation for attacking cancer by targeting the TOR pathway. In fact, almost a year ago, the first anti-cancer mTOR-inhibitor received FDA approval. It's Toricel (temsirolimus), an intravenous drug from Wyeth Pharmaceuticals, for kidney cancer. Novartis has an oral drug (everolimus) for kidney cancer in Phase III trials. (It's already been approved by the FDA as an immunosuppressant to prevent rejection of organ transplants.) Interestingly, and unsurprisingly, everolimus is a derivative of Rapamycin (sirolimus) – an anti-fungal and immunosuppressive compound – which led to the original discovery of mTOR. Everolimus works similarly to Rapamycin as an mTOR inhibitor.

The American biotech company Ariad Pharmaceuticals has a small molecule anti-cancer mTOR inhibitor called deforolimus in intermediate clinical trials for a variety of solid cancers, such as sarcomas, endometrial, prostate, breast and non-small cell lung cancers. The company describes the drug as "a novel small-molecule inhibitor of the protein mTOR, a “master switch” in cancer cells. Blocking mTOR creates a starvation-like effect in cancer cells by interfering with cell growth, division, metabolism, and angiogenesis." Last summer Ariad entered into a major partnership with Merck to develop and test the drug, so this is an indication that the drug has definite promise.

Ariad has a nice video you can download, which explains a bit about how their drug works, and about TOR signaling in general. I highly recommend having a look at it, since it covers upstream signals that activate mTOR (growth factors, amino acids, oxygen, energy), downstream effects (synthesis of proteins for cell growth, cell division, metabolism, and angiogenesis). It notes that certain other signaling proteins (PTEN, Akt, PI3K) cause overactivation of mTOR, and it points out that mTOR stimulates the production of the cyclin D cell division protein.

Tags: ,

Calorie restriction, TOR signaling, and aging

Now that I've given some pointers to information about how TOR signaling is involved with metabolism (see here), it seems like an opportune time to mention a recent research announcement in this general area.

How Dietary Restriction Slows Down Aging (4/17/08)
University of Washington scientists have uncovered details about the mechanisms through which dietary restriction slows the aging process. Working in yeast cells, the researchers have linked ribosomes, the protein-making factories in living cells, and Gcn4, a specialized protein that aids in the expression of genetic information, to the pathways related to dietary response and aging.

Here's the key background:
Previous research has shown that the lifespan-extending properties of dietary restriction are mediated in part by reduced signaling through TOR, an enzyme involved in many vital operations in a cell. When an organism has less TOR signaling in response to dietary restriction, one side effect is that the organism also decreases the rate at which it makes new proteins, a process called translation.

The researchers investigated various strains of yeast cells that had low rates of protein production, but increased lifespan. They found that a common characteristic of such cells was mutations to one part of the cell's ribosomes, the complex of RNA and certain proteins which manufactures all new proteins in the cell. The result of these ribosome changes was a decrease in the production of most proteins, except for one, called Gcn4, a transcription factor, whose production increased. The effect seems to depend on the same pathway affected by reduced TOR signaling. Gcn4 is associated with control of amino acid synthesis, and is activated when a cell is starved for amino acids.
To make the link between Gcn4 and longevity, the scientists then asked whether preventing the increase of Gcn4 would block life span extension. In every case, cells lacking Gcn4 did not respond as strongly as Gcn4-positive cells.

"The increased production of Gcn4 in long-lived yeast strains, combined with the requirement of Gcn4 for full life-span extension, makes a compelling case for Gcn4 as an important downstream factor in this longevity pathway," Kaeberlein said.

One might speculate that increased Gcn4 production somehow helps the cell cope with lack of nutrients, and one effect is that the cell takes steps to conserve its resources and slow the rate of aging.

Since reduction of TOR signaling is one way to bring about this effect, TOR inhibitors might help slow aging and increase lifespan, at least in yeast. However, since TOR affects so many other cell functions, the chance for harmful side effects of reduced TOR signaling is high.
"The role of TOR and translation in aging is known to be conserved across many different species, so it's plausible that this function of Gcn4 is conserved as well," Kennedy said. Future research will be aimed at testing this hypothesis.

"Clearly TOR signaling is one component, and perhaps the major component, of the beneficial health effects associated with dietary restriction," said Kaeberlein. "The difficulty with TOR as a therapeutic target, however, is the potential for negative side effects. As we learn more of the mechanistic details behind how TOR regulates aging, we will hopefully be able to identify even better targets for treating age-associated diseases in people."

Tags: ,

Cancer, metabolism, and oncogenes

I want to call attention (somewhat belatedly) to a series of three very good tutorial blog posts at The Daily Transcript. Although they are nominally about changing views regarding cancer and its causes, they actually provide a nice overview of a number of important topics in molecular biology. Reading these posts will be a big help in understanding a lot of things written about here, in particular topics such as:

  • cancer, and how it is "caused" by various factors like metabolism and genetic mutations, and indirectly affected by other biological systems like the immune system
  • metabolism in general, and how problems with metabolism lead to disease conditions like diabetes and metabolic syndrome, perhaps even Alzheimer's disease
  • calorie restriction, and how it seems to play a role in longevity
  • stem cells – what makes them special, how they function biologically and may play a role in the process of cancer
  • important processes in cell biology, such as apoptosis, autophagy, and (of course) the cell cycle itself
  • general topics in molecular biology, such as growth factors, transcription factors, signaling cascades, and cell surface receptors

So here are the links, with a brief summary of each:

From Metabolism to Oncogenes and Back - Part I (3/17/08)
Historical introduction to the subject. Explains how Otto Warbug had the idea, 100 years ago, that the way to understand cancer was through metabolism. Somewhat later, the discovery of the Rous Sarcoma Virus (1916), and much later, after the revolutionary understanding of DNA and modern molecular biology came about, the focus shifted to the role of oncogenes, tumor suppressors, and genetic mutations in cancer.

From Metabolism to Oncogenes and Back - Part II (3/21/08)
More detailed look at the molecular biology of cancer, protein signaling pathways in general, and TOR signaling in particular. This part includes a great diagram of some of the more important signaling pathways as far as metabolism and cancer are concerned. Besides TOR, it clearly emphasizes the importance of the MAP kinase Ras, and the phosphoinositide signaling proteins PI3K, PTEN, and AKT.

From Metabolism to Oncogenes and Back - Part III (4/2/08)
An even more technical summary of recent discoveries about metabolism, and the peculiar kind of metabolic activity found in cancer cells. It appears that a type of enzyme called pyruvate kinase, which occurs in various forms, plays a big role in cell metabolism and whether a cell uses available energy for making sugars, fats, or DNA.

Tags: ,

Induced Pluripotent Stem Cells II

Saturday, April 26, 2008

In this article from the April 4 Science, which I mentioned here, several research reports dealing with induced pluripotent stem cells were discussed. One of these I covered in the post I just noted.

Another just as important report apparently has not yet been formally published, but is (at least temporarily) available online since February 14 at Science Express:

Generation of Pluripotent Stem Cells from Adult Mouse Liver and Stomach Cells
Induced pluripotent stem (iPS) cells have been generated from mouse and human fibroblasts by the retroviral transduction of four transcription factors. However, the cell origins and molecular mechanisms of iPS cell induction remain elusive. This report describes the generation of iPS cells from adult mouse hepatocytes and gastric epithelial cells. These iPS cell clones appear to be equivalent to ES cells in gene expression and are competent to generate germ-line chimeras.

It's not surprising that this is significant research, as it's from the same team of Shinya Yamanaka that was the first to report successful creation of induced pluripotent stem cells. (See here.)

So what is this research about? Well, the investigators used the same four transcription factors (Oct3/4, Sox2, Klf4, and c-Myc) as employed in the majority of previous iPS studies. However, instead of applying the transcription factors to fibroblast cells, they were applied to two types of epithelial cells instead.

Fibroblasts are part of a body's connective tissue. They are involved in structure and support for other tissues and contain large amounts of the protein collagen. They do not divide for the most part, and so it is especially significant that it was possible to reprogram them into a stemcell-like state at all.

Epithelial cells, on the other hand, line the inner and outer surfaces of various body structures, including skin and the gastrointestinal tract. Such cells divide more frequently. They have to, in order to replace other cells of the same kind that are exposed to hostile environments. Epithelial cells also tend to be more adherent to other cells, because they more highly express an adherence protein called E-cadherin.

In some sense, then, epithelial cells are a little more like stem cells to begin with, so one might expect better results when attempting to reprogram them.

This expectation seems to have been met. One of the key differences the researchers found is that reprogrammed epithelial cells had less tendency to form cancerous tumors in mice into which they were included. Certainly not an inconsiderable advantage. This characteristic may be related to the finding that c-Myc seems to play a less essential role in reprogramming epithelial cells.

Specifically, reprogramming of epithelial cells was almost as efficient when c-Myc was not used as when it was included with the other three transcription factors. Yet it was not possible to accomplish reprogramming if any of the other three factors was omitted. In contrast, the efficiency of reprogramming fibroblasts dropped by 90% when c-Myc was omitted.

Another intriguing difference was that reprogrammed epithelial cells contained higher levels of expression of β-catenin than reprogrammed fibroblasts did. (You may recall – see here – that β-catenin is an important part of the Wnt signaling pathway.) In this regard, the reprogrammed epithelial cells are more like true embryonic stem cells than reprogrammed fibroblasts are. It's probably not a coincidence that expression of Nanog is stimulated by β-catenin, (see here), since Nanog is considered important for maintaining stem cell pluripotency.

A further advantage of the use of epithelial cells is that many fewer retroviral "integration sites" were needed to include the transcription factor genes into the cell genome, in comparison with fibroblasts. This is another way the risk of cancer is reduced.

Further reading:

Tags: ,

Pluripotency and Lin28

As we discussed here, pluripotent stem cells have been obtained by "reporogramming" various kinds of adult cells. In one case, a set of 4 transcription factors – Oct3/4, Sox2, c-Myc, and Klf4 – were used for the reprogramming. Another research team used a slightly different set – Oct3/4, Sox2, Nanog, and Lin28.

Two of the transcription factors are the same in these two sets. Of those that are different, c-Myc and Nanog are very familiar to molecular biologists for a variety of reasons. (We discussed some of what's known about c-Myc here, and Klf4 is discussed here.)

So what, if anything, is special about Lin28? Quite a lot, it turns out. Apparently Lin28 not only promotes pluripotency, but it also interacts with a very well-known type of microRNA called let-7. As we saw here, let-7 does several things that help suppress cancer. For one thing, let-7 regulates the oncogene Ras, apparently by binding to the mRNA encoding Ras, thereby inhibiting protein expression. (See here.) For another thing, and more to the point, let-7 tends to negate some of the "stemness" of stem cells, and pushes them onto a path for differentiation into more specialized cell types. (See here.) This helps inhibit cancer by reducing the ability of suspected cancer stem cells to proliferate. Let-7 has also been mentioned as an inhibitor of oncogenicity of c-Myc.

Lin28, on the other hand, seems to regulate let-7, and therefore it helps preserve "stemness", but at the same time it may raise the risk for development of cancer. A paper in the April 4 issue of Science describes the research that indicates such activity:

Selective Blockade of MicroRNA Processing by Lin28
Here we show that Lin28, a developmentally regulated RNA binding protein, selectively blocks the processing of pri-let-7 miRNAs in embryonic cells. Using in vitro and in vivo studies, we found that Lin28 is necessary and sufficient for blocking Microprocessor-mediated cleavage of pri-let-7 miRNAs. Our results identify Lin28 as a negative regulator of miRNA biogenesis and suggest that Lin28 may play a central role in blocking miRNA-mediated differentiation in stem cells and in certain cancers.

Some people are suggesting that perhaps at least some "cancer stem cells" are actually more ordinary cancer cells that have been reprogrammed (in part by Lin28) to be capable of more stemcell-like behavior. What's really going on here still seems a bit speculative at this point.

This blog post of 3/25/08 goes into a lot of detail on mircoRNA, let-7, Lin28, and the whole ball of wax: bring 'em all together: cancer, stem cells, miRNAs.

Further reading:

Deconstructing Pluripotency – overview article in April 4, 2008 Science that discusses two stem cell papers, including the one cited above

Let7 miRNAs, Lin-28, Cancer and Stem Cells – 3/24/08 blog post that discusses this research

Lin-28 is Master of Let-7 miRNA Processing – 3/25/08 tongue-in-cheek blog post that discusses the previous blog post and the subject more generally

Tid Bits – 3/28/08 blog post on related topics

Tags: , , ,

Stellar Birth in the Galactic Wilderness

Friday, April 25, 2008

Stellar Birth in the Galactic Wilderness (4/16/08)
A new image from NASA's Galaxy Evolution Explorer shows baby stars sprouting in the backwoods of a galaxy -- a relatively desolate region of space more than 100,000 light-years from the galaxy's bustling center.

The striking image, a composite of ultraviolet data from the Galaxy Evolution Explorer and radio data from the National Science Foundation's Very Large Array in New Mexico, shows the Southern Pinwheel galaxy, also known simply as M83.

In the new view, the main spiral, or stellar, disk of M83 looks like a pink and blue pinwheel, while its outer arms appear to flap away from the galaxy like giant red streamers. It is within these so-called extended galaxy arms that, to the surprise of astronomers, new stars are forming.

M83 – click for 800×800 image

More: here

Comb jellies

Sunday, April 20, 2008

I just thought it was sort of fascinating that comb jellies (also known as Ctenophorae), which aren't true jellyfish but are closely related, are now thought to be the oldest extant linage of the animal kingdom, usurping the title from sponges.

What's especially strange (though hardly inexplicable) is that comb jellies have some features that sponges lack, such as differentiated tissues and a nervous system.

The First Animal On Earth Was Significantly More Complex Than Previously Believed
A new study mapping the evolutionary history of animals indicates that Earth's first animal -- a mysterious creature whose characteristics can only be inferred from fossils and studies of living animals--was probably significantly more complex than previously believed.

Using new high-powered technologies for analyzing massive volumes of genetic data, the study defined the earliest splits at the base of the animal tree of life. ...

Among the study's surprising findings is that the comb jelly split off from other animals and diverged onto its own evolutionary path before the sponge.

How could it be that comb jellies are notably more complex than sponges, in spite of having appeared earlier?
Dunn says that the comb jelly could only have achieved its apparent seniority over the simpler sponge via one of two new evolutionary scenarios:

1. the comb jelly evolved its complexity independently of other animals, after it branched off onto its own evolutionary path; or
2. the sponge evolved its simple form from more complex creatures -- a possibility that underscores the fact that "evolution is not necessarily just a march towards increased complexity," says Dunn. "This scenario would provide a particularly dramatic example of that principle."

The evidence that comb jellies emerged earlier than sponges comes from intensive analysis of genetic data. The general idea is that detailed computer analysis comparing the DNA of different species indicates which species emerged earlier or later. As the researcher explains:
"Even though we looked at fewer than 100 species, they were sampled in such a way that they inform the relationships of major groups of animals relative to each other. Therefore, this study, and others like it, will have implications for the placement of far more species than just those that are sampled."

A somewhat earlier announcement is slightly more specific about what the researchers did:

Tree Of Animal Life Has Branches Rearranged, By Evolutionary Biologists
A study led by Brown University biologist Casey Dunn uses new genomics tools to answer old questions about animal evolution. The study is the most comprehensive animal phylogenomic research project to date, involving 40 million base pairs of new DNA data taken from 29 animal species.

Speaking of jellyfish-like creatures, I highly recommend Jelly Music.

Further reading:

Shock: First Animal on Earth Was Surprisingly Complex – another news report of the research

Phylogenetic fallacies: "early branching equals primitive" – blog post that cautions against certains misinterpretations of genomic data

Genomicron: Phylogenetic fallacies: “early branching equals primitive” – another blog post, elaborating on the preceding one


Stimulating NF-κB for radiation protection

Can't resist posting about this, as it relates to one of my favorite transcription factors, NF-κB. (What' so great about NF-κB? I dunno. Maybe it's the catchy name. Besides, one of the systems it plays an important part in regulating is the immune system, especially with respect to inflammation. It's also important in the connection between inflammation and cancer. For more about it, check some other posts.)

NF-κB is actually a family of proteins, rather than a single one. It is also related to homologous proteins in a wide variety of animals, including fruit flies, sea urchins, anemones, and sponges – which is about as diverse a collection as there is in the animal department.

NF-κB is present in cells most of the time. That is, it doesn't need to await certain cell signaling events in order to be produced, which means it's always ready, albeit in an inactive state, to go to work when needed. For example, when a receptor (IL1R) for the immune system cytokine IL-1 fires up, the resulting signaling activates NF-κB that is already present.

Other receptors related to IL1R, called Toll-like receptors, can also lead to the activation of NF-κB. A specific Toll-like receptor in humans (TLR5) reacts to a bacterial protein called flagellin, which is normally found in the flagella of flagellated bacteria. That's not too surprising – this is one rather obvious way for the immune system to detect the presence of bacteria.

Now, another side of NF-κB is that it also regulates genes that control cell survival and proliferation. Specifically, it causes cells to resist apoptosis. Cancer cells take (unfair) advantage of this fact to upregulate NF-κB in order to promote their own nefarious proliferation. In particular, some cancer cells eventually become resistant to the radiation used in radiotherapy – precisely because those cells are the survivors of earlier radiotherapy.

All this background led some clever biologists to think that NF-κB could be useful in cases when one wants to protect cells from being killed by radiation – such as following a nuclear accident or explosion. It might also be helpful for astronauts on long space missions far from the Earth's protective magnetic shield against radiation.

Cancer researchers are looking for safe ways to inhibit NF-κB from protecting cancer cells. However, other biologists reasoned that doing the opposite might be a good way to protect healty cells from radiation. And further, perhaps exposing cells to flagellin might be a convenient way of activating NF-κB (instead of using an actual bacterial infection).

That line of thinking led to this:

New Drug Protects against Radiation Damage
A new drug may protect healthy tissue during cancer-killing radiation treatments or other exposures. Molecular geneticist Andrei Gudkov and colleagues report in Science this week that they protected mice from the cell-damaging effects of radiation by injecting them with a compound that helps cells resist apoptosis, or self-destruction.

Previous studies have found that cancerous cells use nuclear factor kappa-beta--a transcription factor, or protein that turns on or off a gene's protein-making ability--to outlive normal cells and grow out of control. But healthy cells in the gut switch on the same transcription factor when they interact with benign and beneficial bacteria that reside there. Specifically, the protein flagellin in some of the microorganisms' whiplike tails (which they use for propulsion) binds with a receptor on the gut cell and triggers the production of the transcription factor.

The experiment of injecting mice with flagellin and then exposing them to ordinarily lethal radiation seems to have been successful:
The injection not only protected the mice's cells but also toughened them against the effects of free radicals (molecules that can damage DNA or genetic material inside them) as well as beefed up the animal's immune systems. Mice without the injection died after the radiation treatments. "Never before has a single agent been capable of doing all three things together," [lead researcher Andrei] Gudkov says.

Further reading:

Bacteria tails could protect against 'dirty' bomb – news article in NewScientist

Drug Experiment Blocks Radiation Damage – AP news article – also here

Drug Bestows Radiation Resistance on Mice and Monkeys – news article in Science about the research.

An Agonist of Toll-Like Receptor 5 Has Radioprotective Activity in Mouse and Primate Models – technical paper in Science that reports the research.

Tags: ,

Spitzer Sees Shining Stellar Sphere

Spitzer Sees Shining Stellar Sphere
Millions of clustered stars glisten like an iridescent opal in a new image from NASA's Spitzer Space Telescope.

Called Omega Centauri, this sparkling orb of stars is like a miniature galaxy. It is the biggest and brightest of the more than 150 similar objects, called globular clusters, that orbit around the outside of our Milky Way galaxy. Stargazers at southern latitudes can spot the stellar gem with the naked eye in the constellation Centaurus.

Omega Centauri – click for 800×913 image

More: Omega Centauri Looks Radiant in Infrared
Globular clusters are some of the oldest objects in our universe. Their stars are over 12 billion years old, and, in most cases, formed all at once when the universe was just a toddler. Omega Centauri is unusual in that its stars are of different ages and possess varying levels of metals, or elements heavier than boron. Astronomers say this points to a different origin for Omega Centauri than other globular clusters: they think it might be the core of a dwarf galaxy that was ripped apart and absorbed by our Milky Way long ago.

In fact, as would be appropriate for what was once an independent galaxy, it seems that Omega Centauri may contain its own intermediate-mass black hole:

Black Hole Discovered In Center Of Enigmatic Omega Centauri
A new discovery has resolved some of the mystery surrounding Omega Centauri, the largest and brightest globular cluster in the sky. Images obtained with the Advanced Camera for Surveys onboard the NASA/ESA Hubble Space Telescope and data obtained by the GMOS spectrograph on the Gemini South telescope in Chile show that Omega Centauri appears to harbour an elusive intermediate-mass black hole in its centre.

Consciousness, free will, etc.

Thursday, April 17, 2008

For quite a long time, the subject of "consciousness" was not considered especially suitable for scientific investigation. A tremendous amount has been written about it, of course. But most of that has been written by philosophers rather than working scientists. Quite a few, if not a substantial majority, of philosophers seem to think that consciousness is a "mystery" that isn't likely to be understood scientifically anytime soon, if ever.

Well, on one hand, when a subject is not very amenable to scientific investigation, then it usually is philosophers, rather than scientists, who write and speak about it. Only when you see that situation change, and you have actual experimental scientists working on the problem and having the technology to actually conduct meaningful experiments, is it time to regard the subject as fair game for science.

At one time even physics was a subject discussed more by philosophers than by scientists. But that changed when people like Galileo and Newton came along, followed by many others. In regard to consciousness, perhaps the technology known as "functional MRI" may be the factor that tips the scales towards real science.

On the other hand, my general opinion of philosophers is well expressed by a quote attributed to one of the all time greatest scientists, C. F. Gauss: "When a philosopher says something that is true, then it is trivial. When he says something that is not trivial, then it is false."

However that may be, we are now in fact seeing real scientific work being done on the subject of consciousness. It's a vast field, I can only scratch the surface here, and I won't go into much detail about any of this. But perhaps a good example of relevant recent research on consciousness is this:

Brain scanner predicts your future moves
Long before you decided to read this story, your brain may have already said "click that link".

By scanning the brains of test subjects as they pressed one button or another – though not a computer mouse – researchers pinpointed a signal that divulged the decision about seven seconds before people ever realised their choice. The discovery has implications for mind-reading, and the nature of free will.

More reports on this research: here, here
Much more: here

Seven seconds. Think about that for a bit. Where, exactly, is "free will", if the brain has already made, or almost made, a decision seven seconds before one is even conscious of a decision having been made?

Sure, one can argue that "free will" comes into play near the final stage of the process, when a course of action is consciously considered before commitment is made to it. Perhaps. But consider something like drug addiction. Just how much "free will" does an addict actually have to resist his/her cravings? Rather little, no? There's quite a bit of research into addiction now, and it is supporting the idea that addictive behavior is largely driven by brain chemistry. I could cite dozens of recent reports in this area, but I need to move on to other stuff right now. Maybe more on addiction later.

Oh yes, and there are a number of other brain states that appear to significantly affect behavior much more strongly than does conscious deliberation. States such as pain, extreme hunger or thirst, the experience of being tortured by agents of the U. S. government, etc.

There's another area of research that comes to mind in connection with significant unconscious influences on people's thoughts and behavior. Psychologists have a term for it: "priming". What it means is that one can manipulate, at least on a statistical basis, the behavior of another person, such as an experimental subject, a voter, or a "consumer", by subtly exposing the subject to various kinds of cues before eliciting the behavior to be manipulated. The subject may be consciously aware of the cues, but not of how they affect subsequent behavior.

Again, I could probably come up with many examples of studies on this if I had more time. Maybe later. Just two examples now, this story that appeared last year: Who’s Minding the Mind?. I wrote about that here, along with another, somewhat older story (A New Study Suggests A Relationship Between Fear Of Death And Political Preferences). Something to think about in this election year.

Moving right along, this all sets the stage for the following very recent report:

Blind to Change, Even as It Stares Us in the Face
The phenomenon that Dr. Wolfe’s Pop Art quiz exemplified is known as change blindness: the frequent inability of our visual system to detect alterations to something staring us straight in the face. The changes needn’t be as modest as a switching of paint chips. At the same meeting, held at the Italian Academy for Advanced Studies in America at Columbia University, the audience failed to notice entire stories disappearing from buildings, or the fact that one poor chicken in a field of dancing cartoon hens had suddenly exploded. In an interview, Dr. Wolfe also recalled a series of experiments in which pedestrians giving directions to a Cornell researcher posing as a lost tourist didn’t notice when, midway through the exchange, the sham tourist was replaced by another person altogether.

The article is a report by Natalie Angier on research into consciousness and attention by neuroscientist Jeremy Wolfe. I won't attempt to say much about it, except that it concerns the interplay of consciousness and attention. And how in some sense we are not really conscious of as much as we think we are. The sense data are streaming into our brains, and they do register at some level that appears to be conscious. But in fact, only a part of the information stream that is the focus of "attention" actually seems to matter." Attention" is the brain's mechanism for limiting actual processing to just a part of the data stream that, somehow, seems most important.

I won't attempt to summarize more than that. Angier is a pretty good writer. Read the article for yourself. Actually, speaking of Angier, I should remark that I have expressed reservations before about her writing style (here). There I described her style as "too flowery and gaudy for my taste". The article mentioned above is fairly tame in that regard. But sometimes what she writes seems, to me anyhow, to be best described as somewhat twee.

Now, "twee" is a word you may not be familiar with if you have been exposed mainly to U. S. English (and you don't do crossword puzzles). It's a British word. A rough American equivalent might be "affected". However, I used "twee" to make a point. Namely, that if you didn't know the meaning, but you looked it up, I'm pretty sure you won't have much trouble remembering it. Why? Because I brought it to your attention.

So you see, although you were probably conscious of the word when you read it, having had your attention focused on it will tend to be the catalyst that gets it added to your memory.

The brain has other mechanisms for directing consciousness in certain ways and for raising the probability that certain kinds of data get remembered. Emotion is one such psychological mechanism that has this effect. I've touched on this tangentially when I wrote about stress and memory here. Also think about "flashbulb memories".

The point is that sensory data that is accompanied by significant emotional valence tends to be preferentially stored in memory. Various hormones and growth factors (such as BDNF) seem to play a role in this process. Consciousness per se... not so much. You have been conscious of quite a lot that you have already forgotten the next day (such as, perhaps, what you had for breakfast a day or two ago).

Here's another report of recent research that tends in the same direction. It concerns the role of emotion in the sense of smell, and which olfactory experiences tend to be stored in memory:

One Bad Experience Linked To Sniffing Out The Danger
Each human nose encounters hundreds of thousands of scents in its daily travels perched front and center on our face. Some of these smells are nearly identical, so how do we learn to tell the critical ones apart?

Something bad has to happen. Then the nose becomes a very quick learner.

New research from Northwestern University's Feinberg School of Medicine shows a single negative experience linked to an odor rapidly teaches us to identify that odor and discriminate it from similar ones.

More reports on this research: here, here.

I don't think I need to comment further on that, with respect to how the emotional circuitry of the brain influences the memory storage circuitry.

Time to wrap up now. I'm going to indulge in a bit of anecdotal reporting on a few of my own observations on consciousness. This is just speculation on my part, not scientific data at all, of course. Think about your own experiences and see whether they aren't consistent with the possibility that consciousness is just another brain mechanism, which serves various purposes, and interacts in different ways with other brain mechanisms. If this is so, eventually we should be able to understand scientifically what the biology of consciousness is.

First off, I'll note that I recently had a colonoscopy. (Nothing bad found. Thanks for asking.) The most interesting part of the experience was how readily consciousness can be turned on and off. Before the procedure began, I was injected with two drugs: Demerol and Versed. The former blocks pain. The latter is a strong sedative that basically turns off consciousness for a brief, fairly predictable period of time.

For a couple of minutes after receiving the Versed, I didn't notice anything unusual, no drowsiness, no disorientation, nada. I just looked around and noticed various features of the immediate environment, such as the staff and the monitoring instruments. I can still picture that stuff fairly clearly. Of the next half hour or so while the procedure was going on, I can recall nothing at all. And then I was awake again, and everything seemed pretty normal. I can recall that period clearly too.

What I conclude from this is that consciousness and memory formation are both processes and/or mechanisms that can be turned off pretty mechanically with a simple chemical. Probably the memory formation is turned off first, so that the intermediate state of drowsiness (if it even occurred) was not remembered. And then as soon as the chemical has been metabolized away, consciousness and memory formation resume with little aftereffect. (Physicians maintain that a patient's judgment can be compromised for several hours afterwards. Perhaps so, but I could detect little evidence of that.) I was also told that patients under the effects of the drugs are still consciousness enough to follow verbal instructions. But of course, I have no memory of whether this is true.

This experience was not like falling asleep normally, when there is usually a definite period of drowsiness that one can often remember the next day. Nevertheless, once asleep one is no longer conscious in the usual sense. Again this suggests that consciousness is just a mechanism that can be turned off (though not by means of volition, unfortunately) at appropriate times.

But we all realize that many mental processes do not cease when we are asleep. For instance, perceptual data is still coming in and processed by the brain to some extent. Noises, especially, that can and do wake us up. Then there is dreaming, which seems to engage large parts of the machinery of consciousness, including emotional subsubsystems (fear and pleasure), and perceptual processing systems of vision and hearing – only on internally generated rather than external sensory data. Sometimes external perceptual data becomes part of the dream consciousness, but usually not.

Another interesting aspect of dreaming is how it interacts with memory. We all know how quickly, after we wake up, we forget about what we may have been dreaming. This suggests that the short term memory system has been functioning, but loses its content more or less as usual. Intermediate and long term memory systems seem to be shut down. It's very unusual (in my experience) to remember any dreams several hours later, unless I happened to think of them immediately after waking up.

Only, this doesn't seem to be entirely true. I've not infrequently had the experience with dreams, when I can "remember" details of being in places and situations and in the company of people that I've encountered in other dreams, perhaps not at all recently. So it seems that there is some sort of long term memory mechanism and storage capability that is specifically dedicated for use while dreaming. Perhaps this is just an aspect of the déjà vu effect.

Another intesting question about the state of consciousness during sleep is whether, or to what extent, the brain continues to engage in creative problem solving. We all know about the advice, when one has a difficult problem, to "sleep on it". This does seem to help a little, but probably it's more a case of letting sleep restore the freshness and alertness of one's mind. Ceasing to think about problems for awhile (whether hours or days) often has similar benefits. I'd say that I experience creative insights into problems rather more often in some place like the shower or out on a walk than I do right after awakening. So it doesn't seem to me that a great deal of actual ratiocination is going on during sleep. At least in my experience.

One last sort of observation, concerning how attention facilitates memory. I think most people find it a little difficult to recall what they've had to eat for lunch even a day or two ago, certainly a month or even a week ago. Unless, that is, there was something unusual about the circumstances of the meal. Such as eating something one hasn't tried before, or at least not for a long time. Or having a meal in a restaurant or location one hasn't eaten in often (or ever). In such cases, one tends to recall many little details of the experience, not just the items in the meal itself. For instance, one tends to remember noticing specific ingredients in what's eaten, and what flavors seemed to be especially pleasant or unpleasant.

When I think about it, I can still recall where I was the first time I had a cola beverage, and how I enjoyed it. I was only six or so. Now, perhaps what I can remember was not actually the first time. And perhaps I'm only remembering past experiences of having that memory. But still... this is about an experience that was decades ago, and without particularly intense related emotions, just general pleasure. But the experience was marked by having my full attention at the time. Needless to say, I can't summon up the perceptual experience of (probably) any other consumption of a similar beverage... except the one I had today.

From that experience, I draw the conclusion that attention is a mechanism which is separate from consciousness, yet which regulates it in such a way that essentially permanent long term memories can be formed. I don't think it's too much of a stretch to imagine that specific biological features will eventually be identified that implement the mechanisms of attention and consciousness in general.

Exploding star in NGC 2397

Sunday, April 13, 2008

Exploding star in NGC 2397 (3/31/08)
The latest image from the NASA/ESA Hubble Space Telescope reveals a sharp view of the spiral galaxy NGC 2397. This image also shows a rare Hubble view of the late stages of a supernova - SN 2006bc, discovered in March 2006.

NGC 2397, pictured in this image from Hubble, is a classic spiral galaxy with long prominent dust lanes along the edges of its arms, seen as dark patches and streaks silhouetted against the starlight. Hubble’s exquisite resolution allows the study of individual stars in nearby galaxies.

Located nearly 60 million light-years away from Earth, the galaxy NGC 2397 is typical of most spirals, with mostly older, yellow and red stars in its central portion, while star formation continues in the outer, bluer spiral arms. The brightest of these young, blue stars can be seen individually in this high resolution view from the Hubble’s Advanced Camera for Surveys (ACS).

NGC 2397 – click for 1280×1022 image

More about alternative energy

Saturday, April 12, 2008

About a month ago, I wrote about the shortcomings of various alternative energy sources. That was mainly about a variety of problems with nuclear energy, solar energy (photovoltaics), and hydrogen.

I didn't even get into the subject of biofuels, but I should have, because the problems in that area are becoming painfully obvious.

Ordinarily I would not expect to find much significant reporting on a scientific/technical subject in Time magazine, especially something that challenges "conventional wisdom". But via DarkSyde at Kos I see there's an interesting article on the problems of "biofuel": The Clean Energy Scam
Several new studies show the biofuel boom is doing exactly the opposite of what its proponents intended: it's dramatically accelerating global warming, imperiling the planet in the name of saving it. Corn ethanol, always environmentally suspect, turns out to be environmentally disastrous. Even cellulosic ethanol made from switchgrass, which has been promoted by eco-activists and eco-investors as well as by President Bush as the fuel of the future, looks less green than oil-derived gasoline.

Meanwhile, by diverting grain and oilseed crops from dinner plates to fuel tanks, biofuels are jacking up world food prices and endangering the hungry.

The Time article focuses on the loss of rainforest, and consequently the loss of its ability to soak up and sequester CO2. When the forest is gone, CO2 will still be incorporated in biomass (crops of some sort). But then that is converted to biofuel, and released back into the atmosphere when it's burned. (To say nothing of the energy that's just wasted along with release of CO2 when the forest biomass is burned to clear it away.) Given all the energy that has to be expended to grow and harvest biofuel crops, with resulting additional release of CO2, we are worse off in terms of greenhouse gas emissions than if we just burned oil (or even coal).

But that's not the only serious problem. Crops that are grown to make fuel (from sugar cane, corn, switchgrass, or whatever) use land where food crops (for people and animals) could be grown instead. Driving up the cost of food for everyone on the planet. (Have you checked the price of bread or eggs at the market recently?)

Economists have spoken out about this problem for several years, when the hype for biofuels and ethanol was just beginning to build. For instance, we have from Howard Simons in early 2006: Making Our Food Fuel Isn't the Answer
If high prices strengthen energy's claim on food supplies, governments everywhere will intervene on behalf of their hungry citizens. If low prices torpedo biofuels' economics, governments everywhere will respond with subsidies for these industries. Only an elimination of current mandates and subsidies today will avoid these problems tomorrow, but the likelihood of this happening is near zero. Somehow I believe we will rue the day when we decided to make food and fuel substitutes at the margin.

In early 2007 Paul Krugman picked up the story: The Sum of All Ears
There is a place for ethanol in the world’s energy future — but that place is in the tropics. Brazil has managed to replace a lot of its gasoline consumption with ethanol. But Brazil’s ethanol comes from sugar cane.

In the United States, ethanol comes overwhelmingly from corn, a much less suitable raw material. In fact, corn is such a poor source of ethanol that researchers at the University of Minnesota estimate that converting the entire U.S. corn crop — the sum of all our ears — into ethanol would replace only 12 percent of our gasoline consumption.

So ethanol doesn't even help the U. S. all that much in terms of dependence on foreign oil. And this February Krugman returned to the subject here, linking to this: Ethanol Demand in U.S. Adds to Food, Fertilizer Costs
About 33 percent of U.S. corn will be used for fuel during the next decade, up from 11 percent in 2002, the Agriculture Department estimates. Corn rose 20 percent to a record on the Chicago Board of Trade since Dec. 19, the day President George W. Bush signed a law requiring a fivefold jump in renewable fuels by 2022.

Increased demand for the grain helped boost food prices by 4.9 percent last year, the most since 1990, and will reduce global inventories of corn to the lowest in 24 years, government data show. While advocates say ethanol is cleaner than gasoline, a Princeton University study this month said it causes more environmental harm than fossil fuels.

And then last week Krugman had even more: Grains Gone Wild
The subsidized conversion of crops into fuel was supposed to promote energy independence and help limit global warming. But this promise was, as Time magazine bluntly put it, a “scam.”

This is especially true of corn ethanol: even on optimistic estimates, producing a gallon of ethanol from corn uses most of the energy the gallon contains. But it turns out that even seemingly “good” biofuel policies, like Brazil’s use of ethanol from sugar cane, accelerate the pace of climate change by promoting deforestation.

And meanwhile, land used to grow biofuel feedstock is land not available to grow food, so subsidies to biofuels are a major factor in the food crisis. You might put it this way: people are starving in Africa so that American politicians can court votes in farm states.

Here's a report of a scientific study on the issue: Some Biofuels Risk Biodiversity And Could End Up Harming Environment
Corn-based ethanol is currently the most widely used biofuel in the United States, but it is also the most environmentally damaging among crop-based energy sources.

Finally, to bring this back to a solid scientific foundation, Sean at Cosmic Variance reminds us that Energy Doesn’t Grow on Trees
In particular, biofuels (such as ethanol) and hydrogen are not actually sources of energy — given the vagaries of thermodynamics, it costs more energy to create them than we can get by actually using them, as there will inevitably be some waste heat and entropy produced

Although all this bad news about just about every prospective near-term form of alternative energy is discouraging, there are a few other options that may become available in the slightly more distant future. There's the old perennial, controlled nuclear fusion. Even though work on that is even more active than ever, it's still at least several decades away.

But there's another significant option that's often overlooked: solar power satellites. This technology uses very large arrays of photovoltaic panels high in orbit around the earth. The energy is beamed back to the ground in the form of microwaves. (So this should not be confused with simply using mirrors to redirect additional sunlight, which presents serious problems of its own.)

Solar power satellites also have many uncertainties and potential problems, but the largest is simply boosting enough of them into orbit, and maintaining them. A possible approach to those problems involves space elevators. But those, again, present a whole additional set of challenges.

For now, here are a couple of articles from last fall with more details:

Pentagon backs plan to beam solar power from space

New Space Solar Power Report from DoD NSSO

Tags: , , ,

Transcendental L-functions

Wednesday, April 2, 2008

If you're a mathematician and follow current developments in mathematics, chances are (i. e., non-trivial probability) that you've seen recent news regarding something called "transcendental L-functions". For instance, this:

Glimpses of a new (mathematical) world
A new mathematical object was revealed today during a lecture at the American Institute of Mathematics (AIM). Two researchers from the University of Bristol exhibited the first example of a third degree transcendental L-function. These L-functions encode deep underlying connections between many different areas of mathematics.

The news caused excitement at the AIM workshop attended by 25 of the world's leading analytic number theorists.

There's also a similar press release from the University of Bristol (UK), which is the home institution of the mathematicians who made the discovery: Million-dollar maths prize comes a step closer

The "million-dollar" reference (guaranteed to get the attention of the great unwashed) concerns the prize offered by the Clay Mathematics Institute for a proof (or disproof) of the Riemann Hypothesis.

There have been a handful of blog posts I've come across about this development, such as these:
Unfortunately, none of the above answer the basic question that must be on the mind of anyone who was the least bit interested in the news item to begin with: What the f∗ck is a third degree transcendental L-function?

So far, I've found very little online that answers the question and is even remotely readable by anyone who's not a certificated maven in the field of modern analytic number theory. Including this discussion at the redoubtable n-Category Café. Sadly, the discussion there (counting the comments) consists mostly of plaintive pleas that echo the question stated in the preceding paragraph, and a couple of responses which are very erudite, but quite opaque to all but (at best) a few hundred humans on this planet. (Not that I'm complaining, mind you.)

Happily, I did find one reference, that explains quite nicely, and in a way that's comprehensible to just about anyone who know a modicum of college math, just what a third degree transcendental L-function is (actually, the information seems to come from this PDF):

One L of a discovery

Whilst the Riemann zeta-function itself is now reasonably well understood, its L-function relatives are not. L-functions are analytic continuations of the more general Dirichlet series:

where ai are complex numbers. Just like Riemann's function, L-functions can be represented by infinite products involving the primes. They also satisfy particular functional equations. Functional equations shed light on the properties of those functions that satisfy them, and for L-functions F(s) the functional equation is:

where q is an integer called the level, d is the degree, and the numbers ri are Langland's parameters.

In the functional equation, Γ(s) is the standard Gamma function (which is an analytic continuation to complex numbers of the factorial function). The integer d (the number of Γ(s) factors in the functional equation) is the "degree" in the term "third degree transcendental L-function". If all of the Langland's parameters ri are rational or algebraic numbers, then we have an algebraic L-function. Otherwise if any of the parameters are transcendental, we have a transcendental L-function.

Well, honestly, I suppose all that may still leave more than a few people gasping for breath. If it's still as clear as mud, but you really want to get down and dirty, you could check out the Wikipedia links in the above description.

But in particular, perhaps you'd just like to see a simple example of an L-function. The simplest L-function of all is none other than Riemann's zeta function, which is quite nicely described at Foxmaths in one of the links given earlier.

I'd love to go into all of this in a lot more detail, but that's a project for another time. Instead, I'll just show you what is perhaps the next simplest example of an L-function, namely Dirichlet's L-function. Dirichlet was actually one of Riemann's teachers, and he introduced his L-function in 1837 (just a bit more than 10 years after Riemann was born). Dirichlet was giving a proof of the infinitude of primes in arithmetic progressions. The proof consisted of showing that the infinite product expansion of the L-function diverged for certain values of its argument, and that could happen only if there were infinitely many primes in any particular arithmetic progression.

Here is the definition of the Dirichlet L-function as an infinite series:
L(s,χ) = Σ1≤n<∞ χ(n)/ns

(The summation index runs over positive integers, and is written as shown for convenience in HTML.) In that definition, s is the function argument (a complex number, though Dirichlet considered only the real case), and χ(n) is a function called a "character", which is defined on the integers and takes values that are algebraic numbers. It has the property that χ(mn)=χ(m)χ(n), and it depends on the particular arithmetic progression.

Like the Riemann zeta function (and all other self-respecting L-functions), this L(s,χ) has a functional equation, which relates its value at s to its value at 1-s. The equation is a little messy to state, and it helps if you first define
Λ(s,χ) = (π/k)-(s+a)/2Γ((s+a)/2))L(s,χ)

Here k is an integer that depends on the arithmetic progression, and a is (1-χ(-1))/2 (which must be 0 or 1). Given that, the functional equation is
Λ(1-s,χ*) = (iak1/2/τ(χ))Λ(s,χ)

The superscript * denotes complex conjugation, and τ(χ) is a complex number called a Gauss sum, which depends on χ.

What all this shows is that the Dirichlet L-function is a first degree algebraic L-function. So now you know.

If you've found this note informative, and if you want to read a lot more about algebraic number theory, check out my series of tutorials on the subject. (It's a work in progress, not nearly complete, but it will eventually get into much hairier L-functions.)

Tags: , ,

Factorization of prime ideals in extension fields

Tuesday, April 1, 2008

In this installment of our series on algebraic number theory, we're going to do two things. First, we'll look at how a prime ideal of one ring of algebraic integers factors into multiple prime ideals in the ring of integers of a larger field. This is an absolutely central concern of the theory. We'll look at some simple examples. Second, we'll make some general definitions that are used to describe this factorization. These definitions will be needed in most of what follows.

In order to understand what's discussed here, you'll need to recall a great deal of what's been discussed before. You can find a list of all previous installments here if you need to review.

Suppose E⊇F is a field extension and OE, OF are the corresponding rings of integers. We have OF ⊆ OE, and if I is any ideal of OF then of course I ⊆ OE. I isn't an ideal of OE, because it isn't closed under multiplication by elements of OE, but I⋅OE is an ideal of OE.

If P is a prime ideal of OF, we have no good reason to expect that P⋅OE is prime in OE, and in fact it generally is not. For example, let's look at quadratic extensions of ℚ. So suppose F=ℚ, d is a square-free integer, positive or negative, and d ≠ 0 or 1. Let p∈ℤ be a positive prime. We can ask whether (the ideal corresponding to) p is still a prime ideal of the ring of integers of E=ℚ(√d). That is, we can ask what happens to the prime, principal ideal (p) in Oℚ(√d). (With principal ideals, we can usually get away without specifying what ring they are contained in.)

Consider the Diophantine equation ±p=a2-b2d. If we can solve it by finding a,b∈ℤ that make the equation true, then we can verify that we have a factorization of the ideal (p) in Oℚ(√d) expressed by the equation of principal ideals (p)=(a+b√d)(a-b√d). (Proof: p∈(a+b√d)(a-b√d) since ±p=a2-b2d, so (p)⊆(a+b√d)(a-b√d). But by the definition of a product of ideals, all members of (a+b√d)(a-b√d) are a product of an element of Oℚ(√d) and the number [a+b√d][a-b√d]=±p, and so (a+b√d)(a-b√d)⊆(p).)

So solutions of a certain Diophantine equation tell us about how an ideal (p) of ℤ factors in the integers of a quadratic extension. And in fact, if the equation can be solved, then the prime ideal (p) of ℤ is not also a prime ideal of Oℚ(√d). Is there a converse, that is, can we infer solutions of a Diophantine equation from how ideals factor in extensions? Unfortunately, the situation is more complicated. For instance, if Oℚ(√d) is not a PID, then it is possible for (p) to not be a prime ideal of Oℚ(√d), yet there may be no solutions of ±p=a2-b2d. One of the reasons that algebraic number theory is useful and interesting is that it helps get a handle on the complications, as we shall see.

For simplicity, suppose d≡3 (mod 4). Then we have remarked that the integers of ℚ(√d) are given by Oℚ(√d) = {a+b√d | a,b∈ℤ}. Suppose we could find some integer α∈ℚ(√d) such that the norm Nℚ(√d)/ℚα = ±p. (For short, write Nα for the norm.) So if α=a+b√d with a,b∈ℤ, we have ±p=Nα=a2-b2d. When this happens, we can write ±p=(a+b√d)(a-b√d). Since Nα is a prime (in ℤ), α=a+b√d must be a prime of Oℚ(√d) – and p is not a prime in that ring. To make things really easy for showing examples, let d=3. Trying a few values for a and b, let α=4+√3, so Nα=13=(4+√3)(4-√3). Or if α=8+5√3, then Nα=-11=(8+5√3)(8-5√3). So both 11 and 13 are not primes in Oℚ(√3).

This translates directly to a factorization of the principal ideal (13) in Oℚ(√d), namely (13)=(4+√3)(4-√3). Clearly (13) isn't a prime ideal, since it has two distinct nontrivial prime factors. (11)=(-11) is likewise not a prime ideal. (ℚ(√3) happens to be a PID, but that isn't crucial here.) In general, then, if p∈ℤ is a prime, the principal ideal (p) is a prime ideal of ℤ, but for any square-free d∈ℤ (with d≠0,1), (p) splits into the product of two distinct ideals in the integers of the quadratic extension ℚ(√d) if we can find a,b∈ℤ, with a≠0, such that ±p = N(a±b√d) = a2-b2q. In that case, we have a factorization of (p) in Oℚ(√d) into principal ideals such that (p)=(a+b√d)(a-b√d), and the factors are prime ideals. If a≠0, these will be distinct ideals. (Since p is assumed to be prime in ℤ, if a=0, we must have b=±1, and the two ideals will be the same. In this special case, one actually says that p "ramifies" in the extension field.)

Conversely, what if, for the given d, there are no integers a,b∈ℤ that satisfy the equation ±p = a2-b2d – can we conclude that p doesn't split in Oℚ(√d)? Actually, no, we can't in general. We could if Oℚ(√d) happens to be a PID, since, because d≡3 (mod 4), the factor ideals would have to have the form (a±b√d) for some a,b∈ℤ, which would yield a solution of the equation, contrary to supposition. (And remember, by the general theory of Dedekind rings, the same is true if Oℚ(√d) has unique factorization, since a ring of integers of a field is a Dedekind ring, and therefore is a PID, if and only if it has unique factorization.)

However, and this is a major "however", if Oℚ(√d) isn't a PID, or equivalently if it doesn't have unique factorization of elements, then we could have a factorization of ideals where (p)=AB into prime ideals that are not principal. (The prime ideal factors themselves are uniquely determined, which, do not forget, is always true in Dedekind rings.) In that case, the factorization doesn't necessary give us a solution of ±p = a2-b2q. Thus it's possible to have primes p where (p) splits in Oℚ(√d) even if the equation has no integer solutions. We might even have A=B, in which case (p) ramifies.

So one key thing this discussion illustrates is that there is a close, though not entirely simple, connection between solving simple Diophantine equations (like ±p=a2-b2q) and determining how prime ideals of a subfield split into prime ideals in an extension field. Further, these equations arise from asking what numbers can occur as norms of some integer of an extension field. But the situation is more complicated if the ring of integers of the extension field doesn't have unique factorization (or equivalently, if it's not a principal ideal domain). This is another reason uniqueness of factorization (or the lack of it) is rather important.

To give some idea of where this leads, we'll just say that class field theory, which we will be coming to shortly, originated in trying to deal with such questions, in the form of saying someting about how prime ideals split in extension fields, and also about when nonprincipal ideals can become principal ideals in extension fields. It helps us deal with the complexities that result from failure of unique factorization of numbers, and from the equalivalent untidiness of having ideals that are not principal ideals.

But before proceeding, let's establish some general conventions and terminology. The terminology will be used to keep the upcoming discussion succinct, so unfortunately you'll need to memorize the details or else plan to refer back here if you want to follow beyond this point. As a reward, we'll be able to state some fairly simple rules for when primes of ℤ do or do not split in extension fields of ℚ.

Let's set out the terminology to describe the general possibilities for how prime ideals split in arbitrary extension fields. We continue to suppose E⊇F is a finite extension, of degree n=[E:F], not necessarily Galois. Let P be a prime ideal of OF. P⋅OE is an ideal of OE, so it factors in a unique way as a product of powers of distinct ideals that are prime in OE:
P⋅OE = ∏1≤j≤g Qjej
We say that each Qj is a prime lying above P in E. Since every Qj divides P, each one contains P: Qj⊇P. In fact, P=Qj∩OF for each j.

If we start with a prime ideal P of the integers of an extension of ℚ, then P∩ℤ is a prime ideal (p) of ℤ for some rational prime p, and P lies above (p). Given the factorization of P shown above, each Qj also lies above (p). By general commutative ring theory, since Qj is a prime of OE, the quotient ring OE/Qj is a finite field Fq where q is some power of p. (In Dedekind domains, any prime ideal is a maximal ideal, and for any commutative ring R with identity, the quotient of R by a maximal ideal is a field.) Likewise, since P is prime in OF, OF/P is a finite field Fq′ where q′ is also a power of p (and divides q). Each field OE/Qj is a finite extension of the field OF/P, and we denote its degree by fj. Finally, it turns out that P⋅OE is a vector space of degree n=[E:F] over the field OF/P.

These facts allow one to make the following definitions. The exponent ej is called the ramification index of Qj (relative to the given extension). The degree fj is the inertial degree of Qj. And g is called the decomposition number. All of these numbers are specific to how the prime P of OF splits (or decomposes) in OE. These numbers are the key data one wants to have for each prime P for any given extension E⊇F. A prime P is said to be ramified unless all exponents ej=1, in which case it is unramified. Primes that ramify tend to make life more complicated, but fortunately there are only finitely many for any particular extension. All these numbers are related as follows:
n = [E:F] = ∑1≤j≤g ejfj
If P is unramified and all inertial degrees are 1, then we must have g=n, and one says that P splits completely. This is a fairly unusual circumstance. Although only finitely many primes are ramified, most have some inertia.

If E⊇F is a Galois extension, things are much simpler. In that case, all ej have the same value, say e, and all fk have the same value, say f. (This is because the Galois group G(E/F) acts on the various Qj and permutes them among themselves. Since each σ∈G(E/F) is an automorphism, these ideals are all essentially alike.) Hence [E:F]=efg. So there are exactly g primes lying above P in E, and each has the same ramification index and inertial degree.

In the next installment we'll, look at more examples involving quadratic extensions of ℤ, and state the rules for how primes split. The rules will turn out to be based on the classical law of "quadratic reciprocity".

Tags: , ,

Knowledge Link Suggest

Drug Abuse Online Lottery Games Drug Abuse Health and Fitness Health and Fitness free music downloads Law Reports Wallpapers Artist online shop originals Product Auto Parts Web Developer online shop originals Product Insurance Company Health Life Technology System Home Security Beauty Cosmetic Online Travel Agency Linux Tips Guide Shooting Simulator Gadgets Technology Project Build Houses Payday Loans Health News Information Animals Shop Travel Agency North Owl Network Dentist Healthy Aging asian pretty The Tool Hospital Barbecook Best way to make Money with Forex Trading Green Home Design Online Lottery atozbythesea.eu.org augmentezvosventes.eu.org automobielbedrijf.eu.org aworldofwindows.eu.org aws-orgware.eu.org azregionalmls.eu.org bacimamma.eu.org Business Plan Competition travel channel and rental cars Pharmacy and drugs Criminal Justice Planning Agency Lawyers Association The Best of Insurance http://celebritytidings.eu.org http://livingandfashion.eu.org Academic Advising Audio logo Stock Audio auto repair advice guide Healthy Women Medical Centre busy with business Car Finance Tools Debt Settlement Lawyer crafty stash corner cuentas and colors Cheap Wedding Dresses drug abuse statistics Lifestyle Home Solutions education essay titles finance insurance premium Lawyer and Legal Information guide healthy recipes health problems of children health topics kids home decorating service home improvement project home improvement project management home improvement project planning home insurance morgages house insurance plan house insurance progressive The Savings Bank insurance agency business loans insurance business loans insurance premium loan legacy landscape blocks legacy landscape contractors legacy landscape materials personal insurance loan real estate for commercial Bedroom Furniture Lawyer Requirements Concept Design teach about alcohol abuse technology support autocar the family deal tile flooring ideas tips lifetime fitness tips reduce car insurance quotes Online Jewelry Stores traveller hotel Insurance and Finance Auto and Parts wood flooring ideas the networks board game Adult Learning Theories sexy lingerie tattoo designs wallpaper artist health category insurance financial category Category Online Money Category Home Improvement online business category Category software games and android category marketing advertising category marketing advertising Home Solution Real Estate Financing Lucepedia Sports and Fitness Train Now Business Economy Global auto car loans car loans insurance Health Information how to prevent diabetes Best Credit Cards Cheap Wedding Dresses Small Business Finance Systems Body Health Home Improvement Services Guide Android Apps and Games fashion design Alcoholism Android Phones Board Games Cheap Tickets and Vacation Packages Online Prevent Child Obesity Prevent Premature Ejaculation Labor Lawyers Home Interior Design Ideas plumbers and carpet cleaning service diet and slimming Beautiful Picture Doctors Dental Care music video art BurningPedia Guide Planning Home Improvement Beauty Cosmetic Web Development Company lawyer requirements Books On Children Education security innovation work iPhone and iPad Turkey The Medical School Barbecue Electric The Linux Documentation Project Mental Health Care New Car Buying Guide House Curtains Dentist All System Education iPhone Specifications Cloud Trading Strategy mortgages and loans Pharmaceutical Info Online Application Service Provider The Masterchef Business Service Delivery Arts and Entertainment Pokerstars Real Money The Masterchef Electric Motors Gadgets Technology Casino Zone Small Business Solutions Home Advice Fashion Design Clothing Adventure Games Health and Fitness Hospital Equipment Plan Life Insurance public transportation services Prevent Premature Ejaculation Best Professional Carpet Cleaning Web Hosting Vehicle Emissions Inspection Drug Abuse Collection Travel Agency Medical Herbs color vision casting Aromatherapy Tips community film Cardio Tips Top Best Home Improvements Article Marketing Colon Cleanse Tips Terms and Conditions of Sale Products for Treating Acne Scars Natural Beauty Products Seo Web Marketing Business Prospects Trade Online Tool House Cleaner Bathing Equipment Financial Solutions Credit Loans Business Insurance Best Supplements Online Affiliate Marketing Web Design Jobs Online Business Degrees Travel and Leisure Hotel Business Online Payment Processor Women Management Criminal Law Marketing Product Webdesign Transportation Services The Marketing Business Prospects Trade Online Shop bed furniture free movies online Direct Hardware Supply Free Download PDF Technology Ipad doctors Specials Medical Dictionary Legacy Landscape Bank South Pacific Carpet Cleaner Best Internet Business Seo Web Marketing Doctor Healthy Doctors Diet Program Health Benefits of Drinking Water ups pills drugs Best Supplements dynasty fine jewelry Ceramic Floor Tile Online Business Degrees Property Residential and Commercial Real Estate All About Medical Bankruptcy Prevent Child Obesity Handbag and Shoes Education of Children physiotherapy school Cheap Airline Tickets Healthy Life PET online Interzoo Rehabilitation Center electric hybrid vehicles Baidu Software Center Automotive Advertising Agencies Health and Care Virtual private network Student Education Technosound Wedding Dresses excel archives Classic Car Restoration Defense Audit Banking Travel and holiday Forex Trading Jobs Cheap Car Insurance Opening the Book Safe Chemical Act Presspeople System Software Plan Life Insurance Top Best Home Improvements Cheap Car Insurance Auto Parts and Service Swiss Business Acting in Real Time Fitness Equipment Tips Bodybuilding Tips Search Engine Optimization Cardio Tips Beauty Salon Express The Best Advice Lawyer Advice Associated With Online Payday Spa and Wellness Center Packages Holiday Travel European Countries Tours Small Business Finance Electronic Cigarette Commercial Law Advice on Travel Planning Home and Family All System Education Graphic Visual Photo business between countries Shop Online Financial Security Authority Latest Fashion Trends Photoshop Wallpaper Real Estate Agents Micro Cogeneration Beauty Products Business Travel Insurance International Trucks For Sale Hair Tonic Top Web Design Business Financial Touring Start Vehicle Emissions Inspection Business Service Delivery Insurance and Finance Web Design Business Shopping Online Insurance Processing Systems Electric System Vehicle Blood Medicine European Vehicle Market Pharmaceutical Section Home sale Beauty Therapy Acupuncture Recipes Kitchen Restaurant Hardwood Flooring Resident Services The Masterchef Employment and the Economy Free Web Design Paint Creations Travel Insurance SEO Consulting Services Car Insurance Loans Doctors Dental Care Vehicle Wash Systems Search Engine Marketing Online Education Schools Natural Beauty Site Cheap Wedding Dresses annuaire automobile episiotomy and pregnant Westerly Lawyers Healthcare Travel Vacations Cheap iPhone Specifications The ReaL Estate Event Online Pharmacy and Drugstore Music Video Art CareCredit Cloud Trading Strategy House Cleaning Services Interoperability the talent agency Good Home Design shooting simulator Independent Port Services The Hotel Sax The Digital Money Booking Tickets and Hotels Online Business Loans Department of Pharmacy Law Free Credit Report football development Trending in Entertainment Prison Education Sexy Lingerie Wood Flooring Ideas Dealer Optics and Ophthalmology Car Insurance Online Urbanism and Architecture Electronic money International Conference on Education Dental Care Auto Repair Shop Auto Repair Advice Guide Dental Clinic Online Pharmacy Debt Consolidation Loans Free Software Smartphone Advice on benefits Insurance Providence Real Estate Product Research Analyst Product Research Facilitator Fashion Magazine Substance Abuse Education And Technology Marketing Strategy The Best Payday Advance Incident Report Criminal Accounting Books Caring for Pets Real Estate Financing Banking Software Solutions Financial Solutions Credit Loans Full Service Research Agency Discover our free games Business Prospects Trade Online Restaurant Hood Cleaning how to start a business Women's Health and Beauty Payday Loans Criminal Attorney Lawyers Travel and Tourism Fashion Designer The Food Products Fitness and Sports The Beginner's Guide to Businesses Tips To Make SEO Techniques Online Payday Loans Medical Advocate Englisch Training Online Drivers Education Auto Parts and Service Best hotels cheap sport shoes Dental Work Insurance Company Android Phones Financial Advice Service Mobile phone Free Credit Report  The Hair Tonic Advantage Car Rental Information Civil Law Information Payday Cash Loans Payday Cash Loans Advance Affordable Auto Insurance Effective Quality Services Car seats Product Interactions Specialist Medicina Natural Traveller and Hotel European Investment Bank Vehicle Electrical System Pet Food Shop Online Lawyer Employment Plaza Ballroom Weddings Art School The best form of insurance solutions Advice on benefits Insurance Medical coding Tutor Buying Safe Products Free Online Game Famous Fashion Designers Education in Music Virtual Currency Systems Learn Trading Folktronica music Digital currency Manufacturing Engineering Technology Fashionista The Money System Explained Good Marketing In Business Technology Ipad Real Estate Bubble CarePass for Android Business Direct Marketing Carpet Cleaner Natural Beauty Products Seo Web Marketing Business Prospects Trade Online Tool House Cleaner Bathing Equipment Financial Solutions Credit Loans Business Insurance Best Supplements Online Affiliate Marketing Web Design Jobs Online Business Degrees Travel and Leisure Hotel Property Residential and Commercial Real Estate All About Medical Bankruptcy Prevent Child Obesity Handbag and Shoes Education of Children Women Management Free Online Slots Games Healthy Life PET online Interzoo Sunrise Health and Rehabilitation Center Presspeople System Software Baidu Software Center Scientific Entrepreneurship Automotive Advertising Agencies Health and Care Virtual private network Student Education Technosound Information network energy sources Niche Dating Classic Car Restoration Defense Audit Banking Travel and holiday Safe Chemical Act Leader in internet information Music Electric The Marketing verb translation country island FILM LITERATURE GOLD STATUS A Short History of Movies Sexy Lingerie Classified ads Inspire Home Design Trauma Clinic Finance Investment Business Marketing Acting in Real Time Glasgow Tourism Herbal Medicine Cheap Car Insurance Quotes Health Pregnancy Professional Graphic and New Media Designers Legal Connect Network Generating Money Online Oregon health plan Slow Cooker Recipes Best Internet Business Terms and Conditions of Sale Products for Treating Acne Scars Hardwoods Furniture Green Tea Fitness Equipment Tips Bodybuilding Tips Forex Trading Article Marketing Colon Cleanse Tips Hair Loss Tips List Building Search Engine Optimization Cardio Tips Aromatherapy Tips Medical Weight Loss Clinic The Benefits Of Moving Insurance Travel Advice Online Dating Security Network Information Green Energy Scheme Group Auto International Business Telecommunication International All Medicine Reviews Marketplace Real Estate Review Clothing Style Management Consulting Real estate Natural Medicine Treatment Natural Beauty Skin Care Products Shop online Marketing Bank rent car guide Fishing Games cheap website traffic Health and Beauty Gambling football Clothes and Accessories Garden and Landscaping Portal Game Online E-commerce Business Car Accident Lawyer Criminal Law Best free Android apps Hotel and Travel Jobs Technology Film Industry Free Music Downloads Solar Installation Search Engine Optimization Life Technologies Football Betting Site Substance Abuse Apartments home improvement contractor job Luxury Car News Law Products and Services Business Web Hosting Company Complaints Diabetes Drug Abuse Marketing Product Make Money Online Living Without Money Apartments Luxury Car News Law Products and Services Growth Technologies Articles on business growth technologies information Health Insurance Management Advisors System Solar Panel Online Trading The Best of Insurance Payday Loans for Bad Credit Financial Solutions Credit Loans Ensure Insurance Plan Financial Management Team Payday Loans Reviews Distributed File System Student Loan Payments Cheap Insurance in Los Angeles Favorite Fashion Styles The Best Savings Rates Manage Student Loans Webshop europe business guide Shop Personal Loans debt consolidation loans Personal Loan Repayment Chiropractic Tables Webdesign Management Concepts Android Software Free Internet Solution the online casino Best Small Business Blogs Sound Beginnings Automotive Electrical Systems Skin Care in Centrum Master Ethique Cher Client Agency Law Sell Products Online Mens Shoes Strategy Business Development Women's shoes Headache Causes Shop Online Clothing Online Pharmacy Healthy Aging Battle Games best electric vehicle Commercial Real Estate Cosmetic Beauty Product Health Medicine Pharmacy Image of the Orient Fashion Jewellery

Blog Archive