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Philosophia Naturalis #9

Thursday, April 26, 2007

I'm going to try something a little different this time. There have been (as usual) some interesting news stories in the physical sciences recently, and I'm going to start by featuring blog articles that discuss them. So let's get to work.

Extrasolar planets

Undoubtedly, if judged by number of blog posts, the most popular recent news in the physical sciences has concerned extrasolar planets.

And within this category, the biggest buzz has centered on the announcment of the first discovery of an Earthlike planet, orbiting the red dwarf star Gliese 581. Some of the better articles on this are Planet in the Zone at Dynamics of Cats, Another Earth? at Asymptotia, First possibly Earthlike extrasolar planet found! at Bad Astronomy, A Potentially Habitable Earth-like World at Centauri Dreams and "Habitable planet"? Maybe not! at Astroprof's Page.

As if those (and other references they cite) weren't enough, special attention should be called to this and this at Systemic, because the whole blog is about extrasolar planets and contains additional discussion of the discovery at Gliese 581. No word yet, however, on when the first tourist flights will be offered.

Next up is the detection of water vapor in the atmosphere of a distinctly un-Earthlike gas giant planet called HD 209458b. Centauri Dreams, again, has the story on that: Water Vapor in an Exoplanet’s Atmosphere. This followed close on the heels of the announcement about a month before of the detection of any type of molecule in an exoplanet atmosphere – specifically, HD 209458b and another (HD 189733b). Atmosphere of Exoplanets at Blog Physica has some details. Ironically, it was first thought that water was (surprisingly) not present.

And last but not least, an extensive survey has found that almost 30 out of 69 binary systems between 50 and 200 light years from Earth have at least the disks consisting of dust and debris out of which planets could form – though actual planets have yet to be confirmed. See Multiple Suns? at Astroprof's Page and Double Stars May Be Aswarm with Planets at Centauri Dreams for details.

Neutrino oscillation

This news item is about something that wasn't discovered, and on the whole it might be considered a non-event. However... what wasn't discovered was an experimental violation of the standard model of particle physics. Just the latest in a long string of "failures". Specifically, several years ago an experiment turned up possible evidence that in addition to the three known varieties of neutrinos (corresponding to electrons, muons, and tau particles) there might be others, which hardly interact with anything else at all, and hence were called "sterile" neutrinos. Such neutrinos would be completely outside the standard model. Further efforts have been made to confirm this result... and they found nothing. Yet this "failure" is still noteworthy in itself, as yet another case of being unable to find something specific wrong with the standard model, even though just about everyone believes the model is incomplete.

MiniBooNE Neutrino Result, posted by one of the experimenters, Heather Ray, at Cosmic Variance, gives a thorough account. Additional accounts: The Unsinkable Standard Model at Uncertain Principles, MiniBooNE for Neutrinos at Asymptotia, and Working Blind at Charm &c.

Quantum mechanics

QM also continues to be perplexing to just about everyone, because it's so "unreal". Here "reality" is a technical term that refers to definite properties a system might have even though they cannot be measured directly. (This is sometimes called a "hidden variables" theory.) Einstein hated the idea, implicit in the leading interpretations of QM, that this kind of "reality" was, well, an illusion. But all experiments to date point towards that being the case.

QM Says Goodbye to Reality? at Physics and Physicists calls attention to the latest finding of Anton Zeilinger's group in Austria that further disconfirms the notion of "reality".

As Niels Bohr said, "Anyone who is not shocked by quantum theory has not understood it." Another recent post at Physics and Physicists has some thoughts on that: No One Understands Quantum Mechanics?, as a follow-on to the earlier Why is Quantum Mechanics SO Difficult?

Gamma-ray bursts

There was a brief period of a few months last year when astronomers concerned with gamma-ray bursts thought they had these things largely figured out. GRBs came in just two kinds, and there were fairly good models for both kinds. Or so they thought. But nature continues to surprise. New cases keep turning up that don't quite fit previous models. But that's OK, really. Just some additional circumstances that can produce bursts of gamma-rays in ways that haven't (yet) been anticipated.

Dirk Grupe, writing at Scitizen, discusses the most recent example: Gamma-Ray Burst Afterglow That Challenges Gamma-Ray Burst Theory.

Meanwhile, the 11D Space Surfer at Quasar9 writes on the relation of GRBs to an exotic type of neutron star known as a magnetar: GRBs & Magnetars.


And finally, everyone's favorite category: everything else.

Mollishka at A Geocentric View tells us about her recently published research: Variable Stars Near the Galactic Center.

For readers with a philosophical bent, Ponder Stibbons at The truth makes me fret ponders a recent paper of Max Tegmark questioning the validity of the traditional distinction between initial conditions and fundamental laws of physics: Eliminating Initial Conditions — Or Not. And a sequel just showed up today: Confusing Baggage.

As part of an edifying tutorial on concepts of special relativity, Richard Baker at Sharp Blue explains Spacetime and coordinates to us.

Besides GRBs and magnetars, Quasar9 is also interested in the mysteries of black holes: Black Hole (Paradox).

For more along the lines of philosophical musing, Clifford Johnson at Asymptotia offers some observations on an interview with Brian Greene about string theory and suchlike, and in particular with the concept of a "theory of everything": Questions and Answers about Theories of Everything.

And finally, still in the philosophical vein, CuriousCat at The Old Curiosity Shop ruminates on Irreversibility.


That wraps it up for this month. PN will be back again next month, on May 24. There's some quantum uncertainty about its precise location (Δx) in cyberspace, but don't worry. In just Δt we should have more information on that. There'll be an update about it here, and further details here.

Update: As promised, here's the scoop on the next edition of PN: It will be hosted by Stuart Coleman at Daily Irreverence on May 24. Watch for a message there about how to suggest an article for the carnival – Stuart and I both hope you'll help us out with some great suggestions.

Hubble's View of Barred Spiral Galaxy NGC 1672

Monday, April 23, 2007

Hubble's View of Barred Spiral Galaxy NGC 1672
This NASA Hubble Space Telescope view of the nearby barred spiral galaxy NGC 1672 unveils details in the galaxy’s star-forming clouds and dark bands of interstellar dust. NGC 1672 is more than 60 million light-years away in the direction of the southern constellation Dorado. These observations of NGC 1672 were taken with Hubble’s Advanced Camera for Surveys in August of 2005.

NGC 1672 – click for 1000×718 image

More information: here, here.

The Purple Rose of Virgo

Friday, April 20, 2007

The Purple Rose of Virgo (3/27/07)
Until now NGC 5584 was just one galaxy among many others, located to the West of the Virgo Cluster. Known only as a number in galaxy surveys, its sheer beauty is now revealed in all its glory in a new VLT image. Since 1 March, this purple cosmic rose also holds the brightest stellar explosion of the year, known as SN 2007af.

Located about 75 million light years away towards the constellation Virgo ('the Virgin'), NGC 5584 is a galaxy slightly smaller than the Milky Way. It belongs, however, to the same category: both are barred spirals.

NGC 5584 – click for 1280×1024 image

Addiction to tanning?

Wednesday, April 11, 2007

Summer is coming in the northern hemisphere, which means, perhaps, more time in the sun. We had a look here at the surprising role that the anti-cancer protein p53 plays in tanning. Beyond that, we know that getting a tan feels good, if not overdone (so to speak).

(And by the way, studies like this one have shown that vitamin D, which is a byproduct of tanning, has beneficial anti-cancer effects for breast and colorectal cancer, in spite of the risk of melanoma from too much UV exposure. Other studies show protective effects of vitamin D for ovarian cancer, pancreatic cancer, and prostate cancer too.)

Anyhow, it seems that some people can't get enough sun tanning, in spite of the risks. Perhaps it's addictive:

New Study Indicates Tanning May Be Addictive
Despite repeated health warnings about the dangers of tanning from sunlight and artificial light sources, there are still those whose mantra “bronzed is beautiful” remains unshaken. Dermatologists have long suspected that some people may be addicted to tanning – similar to addictions to drugs or alcohol – and refuse to alter their behaviors, even knowing they have an increased risk of developing skin cancer. Now, a new study of college co-eds indicates that some people may be addicted to ultraviolet (UV) light.

Why would this be? Most likely for the same reason that the process of getting a tan feels good. Dermatologist Robert Hornung, who led a questionnaire study to investigate the motivations of dedicated tanners, explains:
“We also know from previous experiments that UV light causes endorphin release, similar to the euphoric sensation associated with intense exercise commonly referred to as ‘runner’s high’ or other pleasure-seeking behavior. Our study set out to find whether certain individuals, particularly those who classify themselves as frequent tanners, exhibit addictive behaviors toward tanning.”

However, it's not clear from what's reported here whether frequent tanners have an actual chemical dependency to their own endorphins. For instance, do they have withdrawal symptoms if they stop tanning abruptly? Are there other signs of a biochemical effect?

Perhaps its time to head to Maui to do a little more research... Or perhaps the nearest nudist resort would be a good choice for some field work. I'd volunteer. Wonder where to apply for a grant...

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Carnival of Mathematics, Ordinal 5

Thursday, April 5, 2007

This edition of the Carnival of Mathematics is dedicated to the memory of Paul J. Cohen (April 2, 1934 - March 23, 2007).

Ironically, he died just 2 weeks ago, the date of the previous edition of this Carnival. Here's the New York Times obituary. There have already been a number of comments about and tributes to Paul in math blogs. One of the shortest, but most significant is from fellow Field's Medalist Terence Tao, who points out that Paul excelled not only in the area of set theory (in which his work best known to the public was done), but also in harmonic analysis.

In fact, Paul was one of the most universal mathematicians (a lot like Tao himself) of the last 50 years. I can personally vouch for that, because he was a Professor in the math department where I did my graduate work. He taught the first year graduate course in complex analysis, and made the subject absolutely inspiring, even though it was new to me at the time, and I was more into algebra than analysis. A year or two later, he conducted a seminar on class field theory (a form of advanced algebraic number theory), and was equally inspiring there. I had the chance to see him in action also in courses or seminars on such subjects as analytic number theory and quadratic forms.

But one thing that impressed me just as much was what a decent, friendly, approachable person he was. Unlike a number of other high-powered mathematicians on the faculty at that time, who were pompous and overbearing. (I won't name names.)

Here's a tribute from another logician, Barkley Rosser, who gives a semi-technical account of the work for which Paul received the Fields Medal. And Jason Rosenhouse at Evolutionblog gives another semi-technical explanation of Cohen's work on the independence from ZFC set theory of the Axiom of Choice and the Continuum Hypothesis.

Paul's set theory work actually settled one of the famous Hilbert Problems, enuciated by David Hilbert in 1900 – the very first, in fact, concerning Georg Cantor's question about the cardinality of the continuum. Was it ℵ1, or indeed any of the ℵs? Kurt Gödel had earlier shown that the answer was "maybe". What Cohen showed was that the answer was "maybe not". It is a measure of how bright this guy was that set theory and logic were not even specialties of his at the time he took on the problem. According to Ben Yandell's book on Hilbert's problems (The Honors Class), Cohen had no training as a logician, but in 1959 he was looking for a challenging new problem to work on, asked logician Solomon Fefferman for some suggestions, and was told about the continuum problem. By 1963 Cohen had cracked the problem in an entirely original way.

Moving on. There has been a lot more news about mathematicians and mathematics that has appeared in the public media recently – somewhat of an anomaly. One item that received relatively light (but non-null) coverage, because of its very technical nature, is the solution of a problem concerning "mock thera functions", originating in some cryptic notes of another mathematical prodigy, Srinivasa Ramanujan. You can read accounts of this work here, here, here, here, and here. Although I haven't come across any blog articles yet that go into much more detail on this, there is a good, detailed post about the fascinating Ramanujan himself from M. Balamurugan's blog.

Another story that got a bit more attention was the discovery that some ornamental art found on Islamic architecture of the 13th century (CE) and later has striking affinities to "quasicrystals" and "Penrose tilings". Two of the better articles were done by Philip Ball and Julie Rehmeyer. More stories about this can be found here, here, here, here, here, here, and here. But the most fascinating aspect of this to me is not the Penrose connection, but instead the connection with "noncommutative geometry", as explained by Masoud Khalkhali at (what else?) the Noncommutative Geometry blog.

Of course, the other big piece of mathematical news recently was the determination of the structure of the exceptional (a technical term, not an encomium) Lie group E8. There have been a number of news reports and blog reports about the work, but this one is interesting, as it comes from a physicist, Clifford Johnson at Asymptotia. That's appropriate, as Clifford is a string theorist, and E8 plays a big role in string theory, with heterotic strings in particular. Interestingly, Clifford notes that E8 is also connected with Penrose tilings.

On the other hand, Peter Woit (a mathematician) at Not Even Wrong is not only skeptical of string theory (to put it mildly), but considers all the publicity surrounding E8 to be a little excessive.

Nevertheless, if you're interested in learning a bit about the mathematics behind E8, you should certainly take a look at two postings from John Armstrong at The Unapologetic Mathematicianhere and here. And if that still isn't enough to satisfy your curiosity, try this wonkier post on Lie groups, Lie algebras, and representations.

By the way, the reason E8 is called an "exceptional" Lie group is not because of some particularly noteworth properties it possesses. The actual reason is that all Lie groups can be understood in terms of a subclass known as "simple" Lie groups. And most of these, in turn, can be classified in terms of several different infinite subclasses. But there are five other simple Lie groups that defy classification – the exceptional Lie groups. And of these, E8 is the largest.

Likewise, we have several additional items to mention here that do not fit into a classification scheme.

If you're a fan of hard problems, there are few problems harder than understanding the Navier-Stokes equations. But we can thank Fields medalist Terry Tao for a lucid explanation of
Why global regularity for Navier-Stokes is hard

If you're more into logical puzzles and computer science, you know all about the "halting problem". Alexandre Borovik at Mathematics under the Microscope uses that topic to illustrate the difference between formal and informal proofs. And incidentally, if you are fond of multi-disciplinary investigations, Alexandre has a book you can download from his site (which is named after the book) that looks at mathematics from the standpoint of cognitive psychology and neuroscience.

My own contribution to this carnival offers a quick (well, compared to a whole book) overview of field theory and Galois theory. It's one step on the way to examining some of the deeper mysteries of algebraic number theory.

Perhaps you've never quite understood why some folks make such a big deal about set theory and the continuum hypotheses, which Paul Cohen did so much to clarify. One illustration of how set theory can actually be applied is to a whole new and elegant way of "constructing" the real numbers. This construction, known as "surreal numbers", was invented by John Horton Conway, another "universal" mathematician. He's also known for, among other things, inventing the "game of life" (based on cellular automata), his role in classifying finite simple groups (only distantly related to Lie groups), and his work (with logician Simon Kochen) on the "free will theorem" in quantum mechanics. Mark Chu-Carroll at Good Math, Bad Math gives us a nice overview of surreal numbers.

Or perhaps you were somehow involved in the "New Math" debacle back in the 70s, and now just don't really care that much for even attempting to explain set theory to junior high students, or practically anyone else for that matter. That's too bad, but math teacher JD2718 has some thoughts on the subject.

In spite of all that, there are rewards to trying to teach math. Dave Marain at MathNotations gives a quick review of the recently published book Coincidence, Chaos, and All That Math Jazz given to him in gratitude by one of his top students. He liked the book.

Thanks, everyone, for reading. Come back again in 2 weeks for the next Carnival of Mathematics, to be hosted by Graeme Taylor at Modulo Errors on April 20.

Note: I had a problem with one of my mailboxes. If you tried to submit an article for this edition of the carnival by sending email to cgd at scienceandreason.net, and the article isn't included here, it may have been affected by the problem. Please resend to carnival at scienceandreason.net, or else submit it for the next edition of the carnival. My apologies for any inconvenience.

Note 2: In the comments Mikael Johanssons observes that his contribution to the carnival got lost in my email problem, so please be sure to check it out. It's about modular representation theory. I'm very sorry about the glitch, Mikael.

Fields and Galois theory

Here's the next installment of our series on algebraic number theory. In the last installment we had a quick look at groups and rings. Now it's time to look at field theory, with special emphasis on what is known as Galois theory. The latter is all about developing a concise description of the relations among the roots of an irreducible polynomial equation using group theory. Some of this theory was famously sketched out in 1832 by Évariste Galois on the night before a duel in which he died.

Galois theory makes it possible to prove several well-known results, such as the impossibility of expressing the solution of some fifth degree polynomial equations in terms of radicals and the impossibility of trisecting some angles with straightedge and compass. We won't go into that, but instead we will eventually see Galois theory used frequently in algebraic number theory.

A field is simply a ring whose multiplication is commutative, has an identity element, and has multiplicative inverses for all elements except the additive identity element. We've already mentioned several examples of fields, specifically number fields, which are algebraic extensions of finite degree of the rationals Q. (I. e., each element of a such a field is an algebraic number in some finite extension of Q.) More exotic examples of fields certainly exist, though, such as finite fields, fields of functions of various kinds, p-adic number fields, and certain other types of local fields. If you go far enough in algebraic number theory, you'll encounter all of these.

The most important set of facts about fields for our purposes lie in what is known as Galois theory. This is the theory developed originally by Évariste Galois to deal (among other things) with the solvability or non-solvability, using radicals, of algebraic equations. It tells us a lot about the structure of field extensions in terms of certain groups – called Galois groups – which are constructed using permutations of roots of a polynomial which determines the extension. (Permutations are 1-to-1 mappings of a set to itself that interchange elements.) A little more precisely, a Galois group consists of automorphisms of a field – i. e. maps (functions) of the field to itself which preserve the field structure. All such automorphisms, it turns out, can be derived from permutations of the roots of a polynomial – under the right conditions.

The importance of Galois theory is that it sketches out some of the "easy" background facts about a given field extension, into which some of the more difficult facts about the algebraic integers of the extension must fit.

Before we proceed, let's review some notations and definitions that will be used frequently. Suppose F is a field. For now, we will assume F is a subset of the complex numbers C, but not necessarily a subset of the real numbers R. If x is an indeterminate (an "unknown"), then F[x] is the set of polynomials in powers of x with coefficients in F. F[x] is obviously a ring. If f(x)∈F[x] is a polynomial, it has degree n if n is the highest power of x in the polynomial. f(x) is monic if the coefficient of its highest power of x is 1. If f(x) has degree n, it is said to be irreducible over F if it is not the product of two (or more) nonconstant polynomials in F[x] having degree less than n.

A complex number α, which is not in F, is algebraic over F if f(α)=0 for some f(x)∈F[x]. f(x) is said to be a minimal polynomial for α over F if f(x) is monic, f(α)=0, and no polynomial g(x) whose degree is less than that of f(x) has g(α)=0. (Note that any polynomial such that f(α)=0 can be made monic without changing its degree.) A minimal polynomial is therefore irreducible over F. F(α) is defined to be the set of all quotients g(α)/h(α) where g(x) and h(x) are in F[x] and h(α)≠0. F(α) is obviously a field, and it is referred to as the field obtained by adjoining α to F.

If E is any field that contains F, such as F(α), the degree of E over F, written [E:F], is the dimension of E as a vector space over F. (Usually this is assumed to be finite, but there are infinite dimensional extensions also.) It is relatively easily proven that if α is algebraic over F and if the minimal polynomial of α has degree n, then [F(α):F]=n. Of course, more than one element can be adjoined to form an extension. For instance, with two elements α and β we write F(α,β), which means (F(α))(β). (Or (F(β))(α) – the order doesn't matter.)

We will frequently need one more important fact. Suppose we have two successive extensions, involving three fields, say D⊇E⊇F. This is called a tower of fields. Then D is a vector space over E, as is E over F. From basic linear algebra, D is also a vector space over F, and vector space dimensions multiply. Consequently, in this situation we have the rule that degrees of field extensions multiply in towers: [D:F]=[D:E][E:F].

Now we're almost ready to define a group, called the Galois group, corresponding to an extension field E⊇F. However, Galois groups can't be properly defined for all field extensions E⊇F. The extension must have a certain property. Here is the problem: The group we want should be a group of permutations on a certain set – the set of all roots of a polynomial equation. But consider this equation: x3-2=0. One root of this equation is the (real) cube root of 2, 21/3. The other two roots are ω21/3 and ω221/3 where ω=(-1+√-3)/2. You can check that ω3=1 and ω satisfies the second degree equation x2+x+1=0. ω is called a root of unity, a cube root of unity in particular. (Roots of unity, as we'll see, are very important in algebraic number theory.) Now, the extension field E=Q(21/3) is contained in R, but the other roots of x3-2=0 are complex, so not in the extension E. This means that it isn't possible to find an automorphism of E which permutes the roots of the equation. Hence we can't have the Galois group we need for an extension like E.

The property of an extension E⊇F that we need to have is that for any polynomial f(x)∈F[x] which is irreducible (has no nontrivial factors) over F, if f(x) has one root in E, then all of its roots are in E, and so f(x) splits completely in E, i. e. f(x) splits into linear (first degree) factors in E. An equivalent condition (as it turns out), though seemingly weaker, is that there be even one irreducible f(x)∈F[x] such that f(x) splits completely in E but in no subfield of E. That is, E must be the smallest field containing F in which the irreducible polynomial f(x)∈F[x] splits completely. E is said to be a splitting field of f(x). The factorization can be written
f(x) = ∏1≤i≤n (x - αi)
with all αi∈E, where n is the degree of f(x). (Remember that we are assuming f(x) is monic.) When this is the case, E is generated over F by adjoining all the roots of f(x) to F. In this case it can be shown that the degree [E:F] is the same as the degree of f(x).

An extension that satisfies these conditions is said to be a Galois extension, and it is the kind of extension we need in order to define the Galois group G(E/F). (Sometimes the type of extension just described is called a normal extension, and a further property known as separability is required for a Galois extension. As long as we are dealing with subfields of C, fields are automaticaly separable, so the concepts of Galois and normal are the same in this case.)

Suppose E⊇F isn't a Galois extension. If E is a proper extensions of F (i. e. E≠F), if α∈E but α∉F, and if f(x) is a minimal polynomial for α over F, then the degree [E:F] of the extension is greater than or equal to the degree of f(x). The degrees might not be equal, because all the roots of f(x) must be adjoined to F to obtain a Galois extension, not just a single root. If α is (any) one of the roots, [F(α):F] is equal to the degree of f(x). But this is the degree [E:F] only if α happens to be a primitive element for the extension, so that E=F(α), which isn't usually the case, and certainly isn't if E isn't a Galois extension of F.

In the example above with f(x)=x3-2, we have E = Q(ω,21/3) = Q(ω)(21/3), [Q(ω):Q]=2 and [Q(ω,21/3):Q(ω)]=3, so the degree of the splitting field of f(x) over Q is 6, because degrees multiply. Q(21/3)⊇Q is an example of a field extension that is not Galois. But Q(ω,21/3)⊇Q(ω) is Galois, since f(x) is irreducible over Q(ω) but splits completely in the larger field. Likewise, Q(ω)⊇Q is Galois, and in fact all extensions of degree 2 are Galois. (If f(x)∈Z[X] is a quadratic which is irreducible over Q and has one root in E, then the roots are given by the quadratic formula and involve √d for some d∈Z, so if one is in E, both are.)

We'll come back to this example, but first we'll look at a simpler one to get some idea of how Galois groups work. Consider the two equations x2-2=0 and x2-3=0. The roots of the first are x=±√2, and the roots of the second are x=±√3. We will start from the field Q and adjoin one root of each equation. This yields two different fields: E2=Q(√2) and E3=Q(√3). If we adjoin a root from both equations we get a larger field that contains the others as subfields: E=Q(√2,√3).

Consider the field extension E2Q first. We use the notation G(E2/Q) to denote the Galois group of the extension. In this example, call it G2 for short. We will use Greek letters σ and τ to denote Galois group elements in general. G2 consists of two elements. One of these is the identity (which we denote by "1") which acts on elements of the field E2 but (by definition) leaves them unchanged. This can be symbolized as 1(α)=α for all α∈E2. The action of a Galois group element can be fully determined by how it acts on a generator of the field, meaning √2 in this case. So it is enough to specify that 1(√2) = √2. This Galois group has just one other element σ2, which is defined by σ2(√2)=-√2. An important property that a Galois group must satisfy is that the action of all its elements leaves the base field (Q in this case) unchanged. A Galois group is an example of a group that acts on a set – a very important concept in group theory. But there is an additional requirement on Galois groups: each group element must preserve the structure of the field it acts on. In technical terms, it must be a field automorphism. We'll see the importance of this condition very soon.

As you can probably anticipate, the Galois group G3=G(E3/Q) has elements 1 and σ3 defined by σ3(√3)=-√3. We can now ask: what is the Galois group of the larger extension E⊇Q? It must contain 1, σ2 and σ3. We have to think about how (for instance) σ2 acts on √3. The clever thing about Galois theory is that it's easy to say what this action should be: σ2 should leave √3 unchanged: σ2(√3)=√3. In particular, σ2(√3) cannot be ±√2 The reason is that σ2 leaves the coefficients of x2-3=0 unchanged, and because σ2 is a structure-preserving field automorphism it cannot map something that is a root of that equation (such as √3) to something that is not a root of that equation (±√2).

For any finite group G, the order of the group is the number of distinct elements. We symbolize the order of G by #(G). In Galois theory it is shown that the order of a Galois group is the same as the degree of the corresponding field extension. Symbolically: #(G(E/F))=[E:F]. Basically this is because we can always find a primitive element θ such that E=F(θ), and θ satisfies an equation f(x)=0, where the degree of f(x) is [E:F]. The other n-1 roots of that equation are said to be conjugate roots. We get n automorphisms, the elements of G(E/F), generated from mapping θ to one of its conjugates (or to itself, giving the identity automorphism). Since the degrees of field extensions in towers multiply, so too do the orders of Galois groups in field towers, as long as each extension is Galois. That is, if D⊇E⊇F, where each extension is Galois, then #(G(D/F)) = #(G(D/E))#(G(E/F)). In our example, the degree of the extension is [Q(√2,√3):Q] = [Q(√2,√3):Q(√2)][Q(√2):Q] = 4. So this is also the order of the Galois group G=G(Q(√2,√3)/Q), and therefore we need to find 4 elements.

We've already identified three of the elements (1, σ2 and σ3). It's pretty clear that the remaining element must be a product of group elements: τ=σ2σ3. The product of Galois group elements is just the composition of the elements, which are field automorphisms (which happen to be derived from permutations on roots of equations), and hence they compose like any other function (or permutation). (Composition is just another term for the the function which is the result of applying one function after another.) Because of how σ2 and σ3 are defined, it must be the case that τ(√2)=-√2 and τ(√3)=-√3. Since E⊇Q is generated by √2 and √3, and τ is a field automorphism, we can figure out what τ(α) must be for any other α∈E. For instance, τ(√6)=√6, since √6=√2√3.

(Remember that we specified σ2(√3)=√3. You may have been wondering why we didn't just define the action of σ2 as an element of the full Galois group G=G(E/Q) by σ2(√3)=-√3. Had we done that, σ2 would have been what we found as τ, while the τ we got as the product of σ2 and σ3 would turn out to be the "old" σ2, so the only difference would be a relabeling of group elements.)

For a slightly more complicated example, suppose f(x)=x2+x+1 and g(x)=x3-2, with roots ω and 21/3 respectively, as above. Then in the tower Q(ω,21/3) ⊇ Q(ω) ⊇ Q both the extensions are Galois. (We already saw this isn't so with the tower Q(ω,21/3) ⊇ Q(21/3) ⊇ Q – order matters.) So the full extension E=Q(ω,21/3) ⊇ Q is Galois. Its Galois group G=G(E/Q) has order 6, because 6 is the degree of the whole extension, since the intermediate extensions are of degree 3 and 2 and the degrees of the extensions multiply.

It turns out to be easy to determine the Galois group of this extension, although there are some tedious calculations needed to verify this. So bear with us a moment here. We can define two automorphisms of E that leave Q fixed, as follows. It suffices to specify them on generators of the field. Let one automorphism σ be defined by σ(&omega)=ω2 and σ(21/3)=21/3. Let the other automorphism τ be defined by &tau(21/3)=ω21/3 and τ(ω)=ω. σ and τ are defined to leave elements of Q unchanged. For sums and products elements of E, σ and τ are defined to preserve the field structure, so they really are automorphisms (though, to be rigorous, this should be checked). So σ and τ are elements of the Galois group G=G(E/Q).

We can also see that σ2(ω) = σ(σ(ω)) = σ(ω2) = ω4 = ω, because ω3 = 1. So σ2 is the identity automorphism. (Note that the exponents on σ and τ refer to repeated composition, not ordinary exponentiation, because composition "is" multiplication in the group G.) If we compute τ2 and τ3 in the same way, applied to 21/3, we find that τ2(21/3) = ω221/3, and τ3(21/3) = 21/3, again because ω3 = 1. Thus τ2 isn't the identity automorphism, but τ3 is.

Now let's compute with the composed automorphisms στ and τσ. First, στ(21/3) = σ(ω21/3) = ω221/3. However, τσ(21/3) = τ(21/3) = ω21/3. So we have στ ≠ τσ, because ω≠ω2. Instead, we will find by a similar calculation that στ(21/3) = ω221/3 = τ2σ(21/3). Hence στ = τ2σ. A little more checking will show that 1 (the identity automorphism), σ, τ, τ2, τσ, and στ give a complete list of distinct automorphisms that can be formed from σ and τ. That's just right, because G must be a group of order 6.

In abstract group theory there are only two distinct groups of order 6. (That is, distinct up to an isomorphism, which is a 1-to-1 structure-preserving map between groups that shows they are essentiall the "same" group.) One is the cyclic group of order 6, denoted by C6. This is isomorphic to the direct product of a cyclic group of order two and one of order 3, i. e. the group C2×C3. However, since στ ≠ τσ, G isn't abelian, it cannot be C6, which is abelian. The only other group of order 6 is (up to isomorphism) S3, the group of permutations of three distinct objects, also known as the symmetric group. (An isomorphic group is the dihedral group D3, the group of symmetries of an equilateral triangle.) Since this group is the only nonabelian group of order 6, G(E/Q) must be isomorphic to it.

There's a whole lot more that could be said about Galois theory, but that would take up quite a bit of space, and the intention here is only to give a feel for what it is about. The basic idea to take away is this: A great deal is known about abstract groups and their subgroup structure. Galois theory is a way to "map" extensions of fields to groups and their subgroups in such a way that most of the interesting details about the extension are reflected in details about the groups, and vice versa. The group structure is sensitive to relationships among elements in the subextensions of a Galois extension. In Galois theory it is proven that there is a precise correspondence between subextensions and subgroups of the Galois group.

It thus becomes possible to infer facts about field extensions easily from a knowledge of their Galois groups. One example of the power of this method is that it made possible proving facts that had remained mysterious for hundreds of years – for example, the unsolvability by radicals of general polynomial equations of degree 5 or more, and the impossibility of certain geometric constructions by straightedge and compass alone (trisecting angles, for example).

Galois theory is an absolutely indispensible tool in algebraic number theory. It will come up again and again. We will mention other results in the theory when they are needed.

In the next installment we'll circle back to take a deeper look at ring theory, which is the most basic tool used in algebraic number theory – because there are generalizations of "integers" in an algebraic number field, and they are rings analogous to the familiar ring Z of ordinary integers.

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Philosophia Naturalis #8 is (finally) up

Sunday, April 1, 2007

And you can find it right here.

We apologize for the delay, but there's some really, really good stuff in this edition. Thanks, Sujit.

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