Also available at http://math.ucr.edu/home/baez/week201.html
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
x^n + y^n = z^n
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
ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0
by radicals. In other words, he showed you can't solve this equation
by means of some souped-up version of the quadratic formula that just
involves taking the coefficients a,b,c,d,e,f and adding, subtracting,
multiplying, dividing and taking nth roots.
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 an
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)
where a and b are rational. Notice: if we add, multiply, subtract or
divide two numbers like this, we get another number of this form.
So this is really a field - and it's called Q(sqrt(2)), since we use
round parentheses to denote the result of taking a field and "extending"
it by throwing in some extra numbers.
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)
where a and b are rational. If sqrt(n) is rational then this field is
just Q, which is boring. Otherwise, we call it a "quadratic number field".
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 pi 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)
for short.
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
Where does complex conjugation come from? It comes from the fact that
we get C from R by throwing in a solution of the quadratic equation
x^2 = -1.
We say C is a "quadratic extension of R". But as soon as we throw in one
solution of this equation, we inevitably throw in another, namely its
negative - and there's no way to tell which is which. And complex
conjugation is the symmetry that switches them!
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
x^2 = -1.
By convention, one of them is called "i", and the other is called
"-i". Having made this convention, there's never any problem telling
them apart. But we could reverse our convention and nothing would
go wrong. For example, if the ghost of Galois wafted into your office
one moonless night and wrote "-i" in all your math and physics books
wherever there had been "i", everything in these books would still be true!
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)
where P and Q are polynomials in z with complex coefficients. You can
add, subtract, multiply and divide rational functions and get other
rational functions, so they form a field. And they contain C as a
subfield, because we can think of any complex number as a *constant*
function. So, we can ask about the Galois group of C(z) over C.
What's it like?
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
So, the only symmetries of the field of rational functions that
act as the identity on constant functions are those coming from
fractional transformations, like this:
f |-> fT where fT(z) = f(T(z))
If you don't follow my reasoning here, don't worry - the details aren't
hard to fill in, but they'd be distracting here.
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, SO_0(3,1):
Gal(C(z)/C) = SO_0(3,1).
We've talked about the Galois group of C(z) over C and the Galois group
of C over R. What about the Galois group of C(z) over R? Unsurprisingly,
this is the group of transformations of the Riemann sphere generated by
fractional linear transformations *and* complex conjugation. And physically,
this corresponds to taking the connected component of the Lorentz group
and throwing in *time reversal*! So you see, complex conjugation is related
to time reversal. But I promised not to go into that....
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)
where K is some algebraic number field and k is some subfield of K.
For starters, consider this example:
Gal(Q(sqrt(n))/Q)
where sqrt(n) is irrational. I've already hinted at what this group is!
Q(sqrt(n)) has sqrt(n) in it, so it also has -sqrt(n) in it, and there's
an automorphism that switches these two while leaving all the rational
numbers alone, namely
a + b sqrt(n) |-> a - b sqrt(n) (a,b in Q)
So, we have:
Gal(Q(sqrt(n)))/Q) = Z/2
just like the Galois group of C over R.
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
1^{1/n}
for lack of a better name, and extend the rationals by this number,
we get a field
Q(1^{1/n})
which contains all the nth roots of unity. Since the nth roots of unity
are evenly distributed around the unit circle, this sort of field is called
a "cyclotomic field", for the Greek word for "circle cutting". In fact,
one can apply Galois theory to this field to figure out which regular
n-gons one can construct with a ruler and compass!
But what's the Galois group
Gal(Q(1^{1/n})/Q)
like? Any symmetry in this group must map 1^{1/n} to some root of unity,
say 1^{m/n} - and once you know which one, you completely know the
symmetry. But actually, this symmetry must map 1^{1/n} to some *primitive*
root of unity, so m has to be relatively prime to n. Apart from that,
though, anything goes - so the size of
Gal(Q(1^{1/n})/Q)
is just the number of guys m less than n that are relatively prime to n. And
if you think about it, these numbers relatively prime to n are just the
same as elements of Z/n that have multiplicative inverses! So if you think
some more, you'll see that
Gal(Q(1^{1/n})/Q) = (Z/n)*
where (Z/n)* is the "multiplicative group" of Z/n - that is, the
elements of Z/n that have multiplicative inverses, made into a group
via multiplication!
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) = S_n
where S_n is the group of all permutations of n things.
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) = S_5
Then, show that if we build up an algebraic number field by starting
with Q and repeatedly throwing in nth roots of numbers we've already got,
we just can't get S_5 as its Galois group over the rationals! We've
already seen this in the case where we throw in a square root of n, or
an nth root of 1. The general case is a bit more work. But instead of
giving the details, I'll just mention a good textbook on Galois theory for
beginners:
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
x^3 = 2
has size divisible by 3, while an iterated quadratic extension has a Galois
group whose size is a power of 2. Using the Galois correspondence, we see
there's no way to stuff the former field into the latter.
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 Qbar for short - that is, Q
with a bar over it.
This field Qbar 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(Qbar/Q).
So, for people studying algebraic number fields,
Gal(Qbar/Q)
is like the holy grail. It's the symmetry group of the algebraic numbers,
and the key to how all algebraic number fields sit inside each other!
But alas, this group is devilishly complicated. In fact, it has literally
driven men mad. One of my grad students knows someone who had a breakdown
and went to the mental hospital while trying to understand this group!
(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(Qbar/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
Qcyc, which is gotten by taking the rationals and throwing in *all* nth
roots of unity for *all* n. Since
Gal(Q(1^{1/n})/Q) = (Z/n)*
we know from Galois theory that Gal(Qcyc/Q) must be a big group containing
all the groups (Z/n)* as closed subgroups. It's easy to see that (Z/n)* is
a quotient group of (Z/m)* if m is divisible by n; this lets us take the
"inverse limit" of all the groups (Z/m)* - and that's Gal(Qcyc/Q). This
inverse limit is also the multiplicative group of the ring Z^, the inverse
limit of all the rings Z/n. Z^ is also called the "profinite completion of
the integers", and I urge you to play around with it if you never have!
It's a cute gadget.
In short:
Gal(Qcyc/Q) = Z^*
and if we stay inside Qcyc, 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 Qcyc is to Qbar as homology theory is to homotopy theory: it's all
about *abelian* Galois groups, so it's manageable.
People now use Qcyc as a kind of base camp for further expeditions into
the depths of Qbar. In particular, since
Q is contained in Qcyc and Qcyc is contained in Qbar
we get an exact sequence of Galois groups:
1 -> Gal(Qbar/Qcyc) -> Gal(Qbar/Q) -> Gal(Qcyc/Q) -> 1
So, to understand Gal(Qbar/Q) we need to understand Gal(Qcyc/Q),
Gal(Qbar/Qcyc) and how they fit together! The last two steps are not
so easy. Shafarevich has conjectured that Gal(Qbar/Qcyc) is the
profinite completion of a free group, say F^. This would give
1 -> F^ -> Gal(Qbar/Q) -> Z^* -> 1
but I have no idea how much evidence there is for Shafarevich's conjecture,
or how much people know or guess about this exact sequence.
More recently, Deligne has turned attention to a certain "motivic" version
of Gal(Qbar/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
Quote of the week:
"Paris, 1 June - A deplorable duel yesterday has deprived the exact
sciences of a young man who gave the highest expectations, but whose
celebrated precosity was lately overshadowed by his political activities.
The young Evariste Galois... was fighting with one of his old friends,
a young man like himself, like himself a member of the Society of
Friends of the People, and who was known to have figured equally in
a political trial. It is said that love was the cause of the combat.
The pistol was the chosen weapon of the adversaries, but because of
their old friendship they could not bear to look at one another and
left their decision to blind fate." - Le Precursor, June 4, 1832
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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 before 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 2^n, 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 pi.
Best wishes from wet Jerusalem,
Avinoam Mann
It's true that Abel and Ruffini beat Galois when it came to the quintic;
the details of this history are covered pretty well by Ian Stewart's book,
I think. And, it's quite true that one doesn't need of Galois theory to
solve a bunch of these problems: for example, to show one can't duplicate
the cube, we just need to see that Q(2^{1/3}) has dimension 3 as a vector
space over Q, while quadratic extensions have dimension 2n. My use of the
Galois correspondence to express this in terms of the size of certain Galois
groups was overkill! The real point of Galois theory is that it provides a
unified framework for tackling a wide range of problems.
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