January 10, 2004

This Week's Finds in Mathematical Physics (Week 201)

John Baez

Lately James Dolan and I have been studying number theory. I used to hate this subject: it seemed like a massive waste of time. Newspapers, magazines and even lots of math books seem to celebrate the idea of people slaving away for centuries on puzzles whose only virtue is that they're easy to state but hard to solve. For example: are any odd numbers the sum of all their divisors? Are there infinitely many pairs of primes that differ by 2? Is every even number bigger than 2 a sum of two primes? Are there any positive integer solutions to

xⁿ + yⁿ = zⁿ

for n > 2? My response to all these was: WHO CARES?!

Sure, it's noble to seek knowledge for its own sake. But working on a math problem just because it's hard is like trying to drill a hole in a concrete wall with your nose, just to prove you can! If you succeed, I'll be impressed - but I'll still wonder why you didn't put all that energy into something more interesting.

Now my attitude has changed, because I'm beginning to see that behind these silly hard problems there lurks an actual theory, full of deep ideas and interesting links to other branches of mathematics, including mathematical physics. It just so happens that now and then this theory happens to crack another hard nut.

I'd known for a while that something like this must be true: after all, when Andrew Wiles proved Fermat's Last Theorem, even the newspapers admitted this was just a spinoff of something more important, namely a special case of the Taniyama-Shimura Conjecture. They said this had something to do with elliptic curves and modular forms, which are very nice geometrical things that show up all over in complex analysis and string theory. Unfortunately, the actual statement of this conjecture seemed impenetrable - it didn't resonate with things I understood.

In fact, the Taniyama-Shimura Conjecture is part of a big network of problems that are more interesting but harder to explain than the flashy ones I listed above: problems like the Extended Riemann Hypothesis, the Weil Conjecture (now solved), the Birch-Swinnerton-Dyer Conjecture, and bigger projects like the Langlands Program and developing the theory of "motives". And these problems rest on top of a solid foundation of beautiful stuff that's already known, like Galois theory and class field theory, and stuff about modular forms and L-functions.

As I'm gradually beginning to understand little bits of these things, I'm getting really excited about number theory... so I'm dying to explain some of it! But where to start? I have to start with something basic that underlies all the fancy stuff. Hmm, I think I'll start with Galois theory.

As you may have heard, Galois invented group theory in the process of showing you can't solve the quintic equation

ax5 + bx4 + cx3 + dx2 + ex + f = 0

The basic idea is something like this. In general, a quintic equation has 5 solutions - and there's no "best one", so your formula has got to be a formula for all five. And there's a puzzle: how do you give one formula for five things?

Well, think about the quadratic formula! It has that "plus or minus" in it, which comes from taking a square root. So, it's really a formula for both solutions of the quadratic equation. If there were a formula for the quintic that worked like this, we'd have to get all 5 solutions from different choices of nth roots in this formula.

Galois showed this can't happen. And the way he did it used symmetry! Roughly speaking, he showed that the general quintic equation is completely symmetrical under permuting all 5 solutions, and that this symmetry group - the group of permutations of 5 things - can't be built up from the symmetry groups that arise when you take nth roots.

The moral is this: you can't solve a problem if the answer has some symmetry, and your method of solution doesn't let you write down a correct answer that has this symmetry!

An old example of this principle is the medieval puzzle called "Buridan's Ass". Placed equidistant between two equally good piles of hay, this donkey starves to death because it can't make up its mind which alternative is best. The problem has a symmetry, but the donkey can't go to both bales of hay, so the only symmetrical thing he can do is stand there.

Buridan's ass would also get stuck if you asked it for the solution to the quadratic equation. Galois proof of the unsolvability of the quintic by radicals is just a more sophisticated variation on this theme. (Of course, you can solve the quintic if you strengthen your methods.)

A closely related idea is "Curie's principle", named after Marie's husband Pierre. This says that if your problem has a symmetry and it is a unique solution, the solution must be symmetrical.

For example, if some physical system has rotation symmetry and it has a unique equilibrium state, this state must be rotationally invariant.

Now, in the case of a ferromagnet below its "Curie temperature", the equilibrium state is not rotationally invariant: the little magnetized electrons line up in some specific direction! But this doesn't contradict Curie's principle, since there's not a unique equilibrium state - there are lots, since the electrons can line up in any direction.

Physicists use the term "spontaneous symmetry breaking" when any one solution of a symmetric problem is not symmetrical, but the whole set of them is. This is precisely what happens with the quintic, or even the quadratic equation.

While these general ideas about symmetry apply to problems of all sorts, their application to number theory kicks in when we apply them to fields. A "field" is a gadget where you can add, subtract, multiply and divide by anything nonzero, and a bunch of familiar laws of arithmetic hold, which I won't bore you with here. The three most famous fields are the rational numbers Q, the real numbers R, and the complex numbers C. However, there are lots of other interesting fields.

Number theorists are especially fond of algebraic number fields. An "algebraic number" is a solution to a polynomial equation whose coefficients are rational numbers. You get an "algebraic number field" by taking the field of rational numbers, throwing in finitely many algebraic numbers, and then adding, subtracting, multiplying and dividing them to get more numbers until you've got a field.

For example, we could take the rationals, throw in the square root of 2, and get a field consisting of all numbers of the form

a + b sqrt(2)

More generally, we could throw in the square root of any integer n, and get an algebraic number field called Q(sqrt(n)), consisting of all numbers

a + b sqrt(n)

Even more generally, we could take the rationals and throw in a solution of any quadratic equation with rational coefficients. But it's easy to see that this doesn't give anything beyond fields like Q(sqrt(n)). And that's the real reason we call these the "quadratic number fields".

There are also "cubic number fields", and "quartic number fields", and "quintic number fields", and so on. And others, too, where we throw in solutions to a whole bunch of polynomial equations!

Now, it turns out you can answer lots of classic but rather goofy-sounding number theory puzzles like "which integers are a sum of two squares?" by converting them into questions about algebraic number fields. And the good part is, the resulting questions are connected to all sorts of other topics in math - they're not just glorified mental gymnastics! So, from a modern viewpoint, a bunch of classic number theory puzzles are secretly just tricks to get certain kinds of people interested in algebraic number fields.

But right now I don't want to explain how we can use algebraic number fields to solve classic but goofy-sounding number theory puzzles. In fact, I want to downplay the whole puzzle aspect of number theory.

Instead, I hope you're reeling with horror at thought of this vast complicated wilderness of fields containing Q but contained in C. First there's a huge infinite thicket of algebraic number fields... and then, there's an ever scarier jungle of fields that contain transcendental numbers like π and e! I won't even talk about that jungle, it's so dark and scary. Physicists usually zip straight past this whole wilderness and work with C.

But in fact, if you stop and carefully examine all the algebraic number fields and how they sit inside each other, you'll find some incredibly beautiful patterns. And these patterns are turning out to be related to Feynman diagrams, topological quantum field theory, and so on...

However, before we can talk about all that, we need to understand the basic tool for analyzing how one field fits inside another: Galois theory!

A function from a field to itself that preserves addition, subtraction, multiplication and division is called an "automorphism". It's just a symmetry of the field. But now, suppose we have a field K which contains some smaller field k. Then we define the "Galois group of K over k" to be the group of all automorphisms of K that act as the identity on k. We call this group

Gal(K/k)

The classic example, familiar to all physicists, is the Galois group of the complex numbers, C, over the real numbers, R. This group has two elements: the identity transformation, which leaves everything alone, and complex conjugation, which switches i and -i. Since the only group with 2 elements is Z/2, we have

Gal(C/R) = Z/2

x2 = -1

Note: we know that i and -i are different, but we can't tell which is which! This sounds a bit odd at first. It's a bit hard to explain precisely in ordinary language, which is part of why Galois had to invent group theory. But it's fun to try to explain it in plain English... so let me try. The complex numbers have two solutions to

x2 = -1

Here's another way to think about it. Suppose we meet some extraterrestrials and find that they too have developed the complex numbers by taking the real numbers and adjoining a square root of -1, only they call it "@". Then there would be no way for us to tell if their "@" was our "i" or our "-i". All we can do is choose an arbitrary convention as to which is which.

Of course, if they put their "@" in the lower halfplane when drawing the complex plane, we might feel like calling it "-i"... but here we are secretly making use of a convention for matching their complex plane with ours, and the other convention would work equally well! If they drew their real line vertically in the complex plane, it would be more obvious that we need a convention to match their complex plane with ours, and that there are two conventions for doing this, both perfectly self-consistent.

If you've studied enough physics, this extraterrestrial scenario should remind you of those thought experiments where you're trying to explain to some alien civilization the difference between left and right... by means of radio, say, where you're not allowed to refer to specific objects you both know - so it's cheating to say "imagine you're on Earth looking at the Big Dipper and the handle is pointing down; then Arcturus is to the right."

If the laws of physics didn't distinguish between left and right, you couldn't explain the difference between left and right without "cheating" like this, so the laws of physics would have a symmetry group with two elements: the identity and the transformation that switches left and right. As it turns out, the laws of physics do distinguish between left and right - see "week73" for more on that. But that's another story. My point here is that the Galois group of C over R is a similar sort of thing, but built into the very fabric of mathematics! And that's why complex conjugation is so important.

I could tell you a nice long story about how complex conjugation is related to "charge conjugation" (switching matter and antimatter) and also "time reversal" (switching past and future). But I won't!

Here's another example of a Galois group that physicists should like. Let C(z) be the field of rational functions in one complex variable z - in other words, functions like

f(z) = P(z)/Q(z)

It's the Lorentz group!

To see this, it's best to think of rational functions as functions not on the complex plane but on the "Riemann sphere" - the complex plane together with one extra point, the "point at infinity". The only conformal transformations of the Riemann sphere are "fractional linear transformations":

       az + b
T(z) = ------
       cz + d

f |-> fT                        where fT(z) = f(T(z)) 

The last step is to check that the group of fractional linear transformations is the same as the Lorentz group. You can do this algebraically, but you can also do it geometrically by thinking of the Riemann sphere as the "heavenly sphere": that imaginary sphere the stars look like they're sitting on. The key step is to check this remarkable fact: if you shoot past the earth near the speed of light, the constellations will look distorted by a Lorentz transformation - but if you draw lines connecting the stars, all the angles between these lines will remain the same; only their lengths will get messed up!

Moreover, it's obvious that if you rotate your head, both angles and lengths on the heavenly sphere are preserved. So, any rotation or Lorentz boost gives an angle-preserving transformation of the heavenly sphere - that is, a conformal transformation! And this must be a fractional linear transformation.

Summarizing, the Galois group of C(z) over C is the Lorentz group, or more precisely, its connected component, SO₀(3,1):

Gal(C(z)/C) = SO0(3,1).

I've been talking about Galois groups that physicists should like, but you're probably wondering where the number theory went! Well, it's all part of the same big story. In number theory we're especially interested in Galois groups like

Gal(K/k)

Gal(Q(sqrt(n))/Q)

a + b sqrt(n) |-> a - b sqrt(n)                  (a,b in Q)

Gal(Q(sqrt(n)))/Q) = Z/2

To get some bigger Galois groups, let's take Q and throw in a "primitive nth root of unity". Hmm, I may need to explain what that means. There are n different nth roots of 1 - but unlike the two square roots of -1, these are not all created equal! Only some are "primitive".

For example, among the 4th roots of unity we have 1 and -1, which are actually square roots of unity, and i and -i, which aren't. A "primitive nth root of unity" is an nth root of 1 that's not an kth root for any k < n. If you take all the powers of any primitive nth root of unity, you get all the nth roots of unity. So, if we take some primitive nth root of unity, call it

11/n

Q(11/n)

But what's the Galois group

Gal(Q(11/n)/Q)

Gal(Q(11/n)/Q)

Gal(Q(11/n)/Q) = (Z/n)*

This group can be big, but it's still abelian. Can we get some nonabelian Galois groups from algebraic number fields?

Sure! Let's say you take some polynomial equation with rational coefficients, take all its solutions, throw them into the rationals - and keep adding, subtracting, multiplying and dividing until you get some field K. This K is called the "splitting field" of your polynomial.

But here's the interesting thing: if you pick your polynomial equation at random, the chances are really good that it has n different solutions if the polynomial is of degree n, and that any permutation of these solutions comes from a unique symmetry of the field K. In other words: barring some coincidence, all roots are created equal! So in general we have

Gal(K/Q) = Sn

Sometimes of course the Galois group will be smaller, since our polynomial could have repeated roots or, more subtly, algebraic relations between roots - as in the cyclotomic case we just looked at.

But, we can already start to see how to prove the unsolvability of the general quintic! Pick some random 5th-degree polynomial, let K be its splitting field, and note

Gal(K/Q) = S5

1) Ian Stewart, Galois Theory, 3rd edition, Chapman and Hall, New York, 2004.

Ian Stewart is famous as a popularizer of mathematics, and it shows here - he has nice discussions of the history of the famous problems solved by Galois theory, and a nice demystification of the Galois' famous duel. But, this is a real math textbook - so you can really learn Galois theory from it! Make sure to get the 3rd edition, since it has more examples than the earlier ones.

Having given Ian Stewart the dirty work of explaining Galois theory in the usual way, let me say some things that few people admit in a first course on the subject.

So far, we've looked at examples of a field k contained in some bigger field K, and worked out the group Gal(K/k) consisting of all automorphisms of K that fix everything in k.

But here's the big secret: this has NOTHING TO DO WITH FIELDS! It works for ANY sort of mathematical gadget! If you've got a little gadget k sitting in a big gadget K, you get a "Galois group" Gal(K/k) consisting of symmetries of the big gadget that fix everything in the little one.

But now here's the cool part, which is also very general. Any subgroup of Gal(K/k) gives a gadget containing k and contained in K: namely, the gadget consisting of all the elements of K that are fixed by everything in this subgroup.

And conversely, any gadget containing k and contained in K gives a subgroup of Gal(K/k): namely, the group consisting of all the symmetries of K that fix every element of this gadget.

This was Galois' biggest idea: we call this a GALOIS CORRESPONDENCE. It lets us use group theory to classify gadgets contained in one and containing another. He applied it to fields, but it turns out to be useful much more generally.

Now, it would be great if the Galois corresondence were always a perfect 1-1 correspondence between subgroups of Gal(K/k) and gadgets containing k and contained in K. But, it ain't true. It ain't even true when we're talking about fields!

However, that needn't stop us. For example, we can restrict ourselves to cases when it is true. And this is where the Fundamental Theorem of Galois Theory comes in! It's easiest to state this theorem when k and K are algebraic number fields, so that's what I'll do. In this case, there's a 1-1 correspondence between subgroups of Gal(K/k) and extensions of k contained in K if:

i) K is a "finite" extension of k. In other words, K is a finite-dimensional vector space over k.

ii) K is a "normal" extension of k. In other words, if a polynomial with coefficients in k can't be factored at all in k, but has one root in K, then all its roots are in K.

For general fields we also need another condition, namely that K be a "separable" extension of k. But this is automatic for algebraic number fields, so let's not worry about it.

At this point, if we had time, we could work out a bunch of Galois groups and see a bunch of patterns. Using these, we could see why you can't solve the general quintic using radicals, why you can't trisect the angle or double the cube using ruler-and-compass constructions, and why you can draw a regular pentagon using ruler and compass, but not a regular heptagon. Basically, to prove something is impossible, you just show that some number can't possibly lie in some particular algebraic number field, because it's the root of a polynomial whose splitting field has a Galois group that's "fancier" than the Galois group of that algebraic number field.

For example, ruler-and-compass constructions produce distances that lie in "iterated quadratic extensions" of the rationals - meaning that you just keep throwing in square roots of stuff you've got. Doubling the cube requires getting your hands on the cube root of 2. But the Galois group of the splitting field of

x3 = 2

But you can read about this in any good book on Galois theory, so I'd rather dive right into that thicket I was hinting at earlier: the field of ALL algebraic numbers! The roots of any polynomial with coefficients in this field again lie in this field, so we say this field is "algebraically closed". And since it's the smallest algebraically closed field containing Q, it's called the "algebraic closure of Q", or Q^- for short - that is, Q with a bar over it. (I can't quite draw it here.)

This field Q^- is huge. In particular, it's an infinite-dimensional vector space over Q. So, condition i) in the Fundamental Theorem of Galois Theory doesn't hold. But that's no disaster: when this happens, we just need to put a topology on the group Gal(K/k) and set up the Galois correspondence using closed subgroups of Gal(K/k). Using this trick, every algebraic number field corresponds to some closed subgroup of Gal(Q^-/Q).

So, for people studying algebraic number fields,

                           Gal(Q-/Q)

(There may have been other reasons for his breakdown, too, but as readers of E. T. Bell's book "Men in Mathematics" know, the facts should never get in the way of a good anecdote.)

If Gal(Q^-/Q) were just an infinitely tangled thicket, it wouldn't be so tantalizing. But there are things we can understand about it! To describe these, I'll have to turn up the math level a notch...

First of all, an extension K of a field k is called "abelian" if Gal(K/k) is an abelian group. Abelian extensions of algebraic number fields can be understood using something called class field theory. In particular, the Kronecker-Weber theorem says that every finite abelian extension of Q is contained in a cyclotomic field. So, they all sit inside a field called Q^cyc, which is gotten by taking the rationals and throwing in all nth roots of unity for all n. Since

Gal(Q(11/n)/Q) = (Z/n)*

In short:

Gal(Q^cyc/Q) = Z^{^}*

and if we stay inside Q^cyc, we're in a zone where the pattern of algebraic number fields can be understood. This stuff was worked out by people like Weber, Kronecker, Hilbert and Takagi, with the final keystone, the Artin reciprocity theorem, laid in place by Emil Artin in 1927. In a certain sense Q^cyc is to Q^- as homology theory is to homotopy theory: it's all about abelian Galois groups, so it's manageable.

People now use Q^cyc as a kind of base camp for further expeditions into the depths of Q^-. In particular, since

Q is contained in Qcyc and Qcyc is contained in Q-

1 → Gal(Q-/Qcyc) → Gal(Q-/Q) → Gal(Qcyc/Q) → 1

1 → F^ → Gal(Q-/Q) → Z^* → 1

More recently, Deligne has turned attention to a certain "motivic" version of Gal(Q^-/Q), which is a proalgebraic group scheme. This sort of group has a Lie algebra, which makes it more tractable. And there are a bunch of fascinating conjectures about this Lie algebra is related to the Riemann zeta function at odd numbers, Connes and Kreimer's work on Feynman diagrams, Drinfeld's work on the Grothendieck-Teichmueller group, and more!

I really want to understand this stuff better - right now, it's a complete muddle in my mind. When I do, I will report back to you. For now, though, let me give you some references.

For two very nice but very different introductions to algebraic number fields, try these:

2) H. P. F. Swinnerton-Dyer, A Brief Guide to Algebraic Number Theory, Cambridge U. Press, Cambridge 2001.

3) Juergen Neukirch, Algebraic Number Theory, trans. Norbert Schappacher, Springer, Berlin, 1986.

Both assume you know some Galois theory or at least can fake it. Neukirch's book is good for the all-important analogy between Galois groups and fundamental groups, which I haven't even touched upon here! Swinnerton-Dyer's book has the virtue of brevity, so you can see the forest for the trees. Both have a friendly, slightly chatty style that I like.

For Shafarevich's conjecture, try this:

4) K. Iwasawa, On solvable extensions of algebraic number fields, Ann. Math. 58 (1953) 548-572.

For Deligne's motivic analogue, try this:

5) Pierre Deligne, Le groupe fondamental de la droite projective moins trois points, in Galois Groups over Q, MSRI Publications 16 (1989), 79-313.

This stuff has a lot of relationships to 3d topological quantum field theory, braided monoidal categories, and the like... and it all goes back to the Grothendieck-Teichmueller group. To learn about this group try this book, and especially this article in it:

6) Leila Schneps, The Grothendieck-Teichmuller group: a survey, in The Grothendieck Theory of Dessins D'Enfants, London Math. Society Notes 200, Cambridge U. Press, Cambridge 1994, pp. 183-204.

To hear and watch some online lectures on this material, try:

7) Leila Schneps, The Grothendieck-Teichmuller group and fundamental groups of moduli spaces, MSRI lecture available at http://www.msri.org/publications/ln/msri/1999/vonneumann/schneps/1/

Grothendieck-Teichmuller group and Hopf algebras, MSRI lecture available at http://www.msri.org/publications/ln/msri/1999/vonneumann/schneps/2/

For a quick romp through many mindblowing ideas which touches on this material near the end:

8) Pierre Cartier, A mad day's work: from Grothendieck to Connes and Kontsevich - the evolution of concepts of space and symmetry, Bulletin of the AMS, 38 (2001), 389 - 408. Also available at http://www.ams.org/joursearch/index.html

For even more mindblowing ideas along these lines:

9) Jack Morava, The motivic Thom isomorphism, talk at the Newton Institute, December 2002, also available at math.AT/0306151.

Addendum: I received the following email from Avinoam Mann, which corrects some mistakes I made:

Dear John,

It's very nice that you've come to appreciate the beauties of number theory, and I enjoyed reading your description of Galois theory, but I hope that you would not mind if I ask you not to help spread some common misunderstandings about it. First, it was not Galois who proved the impossibility of solving the quintic by radicals. This was attempted first by Ruffini, I think in 1799, and the proof by Abel, about ten years befors Galois, was the one that the mathematical community accepted. While I often teach Galois theory (e.g. next semester), I never studied Ruffini's and Abel's work in detail. What Galois did was to give a criterion checking for an arbitrary equation whether it is soluble by radicals or not.

Another point: there is no need for Galois theory to prove that duplication of the cube and trisection of an angle cannot be done by ruler and compass. Since ruler and compass constructions are equivalent to solving a series of quadratics, they can lead only to fields F of dimension 2ⁿ, for some n, over the rationals. But the two problems that I mentioned lead to extensions of dimension 3. All this is very elementary. Similar considerations lead to necessary conditions for the constructibility of regular polygons, but proving these conditions sufficient does require more theory (unless, I guess, you provide directly the relevant system of quadratics; I think that is what Gauss did - his proof also preceded Galois). Squaring the circle is, of course, a different matter. Here we need the transcendence of π.

Best wishes from wet Jerusalem,

Avinoam Mann

© 2004 John Baez
baez@math.removethis.ucr.andthis.edu