# January 10, 2004 {#week201}

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 $n$th 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 $n$th 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 $n$th 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 $\mathbb{Q}$, the real numbers $\mathbb{R}$, and the
complex numbers $\mathbb{C}$. 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 $\mathbb{Q}(\sqrt{2})$, since we use round
pare$n$theses 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 $\mathbb{Q}(\sqrt{n})$, consisting of all
numbers
$$a + b \sqrt{n}$$
where $a$ and $b$ are rational. If $\sqrt{n}$ is rational then this field is
just $\mathbb{Q}$, 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 $\mathbb{Q}(\sqrt{n})$. 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 $\mathbb{Q}$ but contained in $\mathbb{C}$. 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 $\mathbb{C}$.

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
$$\mathrm{Gal}(K/k)$$
for short.

The classic example, familiar to all physicists, is the Galois group of
the complex numbers, $\mathbb{C}$, over the real numbers, $\mathbb{R}$. 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 $\mathbb{Z}/2$, we have
$$\mathrm{Gal}(\mathbb{C}/\mathbb{R}) = \mathbb{Z}/2$$
Where does complex conjugation come from? It comes from the fact that we
get $\mathbb{C}$ from $\mathbb{R}$ by throwing in a solution of the quadratic equation
$$x^2 = -1$$
We say $\mathbb{C}$ is a "quadratic extension of $\mathbb{R}$". 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 ["Week 73"](#week73) for
more on that. But that's another story. My point here is that the
Galois group of $\mathbb{C}$ over $\mathbb{R}$ 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 $\mathbb{C}(z)$ be the field of rational functions in one complex variable $z$ ---
in other words, functions like
$$f(z) = \frac{P(z)}{Q(z)}$$
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 $\mathbb{C}$ as a
subfield, because we can think of any complex number as a *constant*
function. So, we can ask about the Galois group of $\mathbb{C}(z)$ over $\mathbb{C}$. 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":
$$T(z) = \frac{az+b}{cz+d}$$
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:
$$f \mapsto fT \qquad \mbox{where $fT(z) = f(T(z))$}$$
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 $\mathbb{C}(z)$ over $\mathbb{C}$ is the Lorentz group, or
more precisely, its connected component, $\mathrm{SO}_0(3,1)$:
$$\mathrm{Gal}(\mathbb{C}(z)/\mathbb{C}) = \mathrm{SO}_0(3,1).$$

We've talked about the Galois group of $\mathbb{C}(z)$ over $\mathbb{C}$ and the Galois group
of $\mathbb{C}$ over $\mathbb{R}$. What about the Galois group of $\mathbb{C}(z)$ over $\mathbb{R}$? 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
$$\mathrm{Gal}(K/k)$$
where $K$ is some algebraic number field and $k$ is some subfield of $K$. For
starters, consider this example:
$$\mathrm{Gal}(\mathbb{Q}(\sqrt{n})/\mathbb{Q})$$
where $\sqrt{n}$ is irrational. I've already hinted at what this group is!
$\mathbb{Q}(\sqrt{n})$ 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
$$a + b \sqrt{n} \mapsto a - b \sqrt{n} \qquad \mbox{($a,b$ in $\mathbb{Q}$)}$$
So, we have:
$$\mathrm{Gal}(\mathbb{Q}(\sqrt{n}))/\mathbb{Q}) = \mathbb{Z}/2$$
just like the Galois group of $\mathbb{C}$ over $\mathbb{R}$.

To get some bigger Galois groups, let's take $\mathbb{Q}$ and throw in a
"primitive $n$th root of unity". Hmm, I may need to explain what that
means. There are $n$ different $n$th 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 $n$th root of unity" is an $n$th root of $1$ that's not an $k$th
root for any $k < n$. If you take all the powers of any primitive $n$th
root of unity, you get *all* the $n$th roots of unity. So, if we take some
primitive $n$th root of unity, call it
$$1^{\frac1n}$$
for lack of a better name, and extend the rationals by this number, we
get a field
$$\mathbb{Q}(1^{\frac1n})$$
which contains all the $n$th roots of unity. Since the $n$th 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
$$\mathrm{Gal}(\mathbb{Q}(1^{\frac1n})/\mathbb{Q})$$
like? Any symmetry in this group must map $1^{\frac1n}$ to some root of unity,
say $1^{\frac{m}{n}}$ --- and once you know which one, you completely know the
symmetry. But actually, this symmetry must map $1^{\frac1n}$ 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
$$\mathrm{Gal}(\mathbb{Q}(1^{\frac1n})/\mathbb{Q})$$
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 $\mathbb{Z}/n$ that have multiplicative inverses! So if you
think some more, you'll see that
$$\mathrm{Gal}(\mathbb{Q}(1^{\frac1n})/\mathbb{Q}) = (\mathbb{Z}/n)^\times$$

where $(\mathbb{Z}/n)^\times$ is the "multiplicative group" of $\mathbb{Z}/n$ --- that is, the
elements of $\mathbb{Z}/n$ 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
$$\mathrm{Gal}(K/\mathbb{Q}) = S_n$$
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
$$\mathrm{Gal}(K/\mathbb{Q}) = S_5$$
Then, show that if we build up an algebraic number field by starting
with $\mathbb{Q}$ and repeatedly throwing in $n$th 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 $n$th 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 $\mathrm{Gal}(K/k)$ 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" $\mathrm{Gal}(K/k)$
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 $\mathrm{Gal}(K/k)$ 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 $\mathrm{Gal}(K/k)$: 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 $\mathrm{Gal}(K/k)$ 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 $\mathrm{Gal}(K/k)$ 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 $\mathbb{Q}$, it's called the "algebraic closure of $\mathbb{Q}$",
or $\overline{\mathbb{Q}}$ for short --- that is, $\mathbb{Q}$ with a bar over it. (I can't quite draw
it here.)

This field $\overline{\mathbb{Q}}$ is huge. In particular, it's an infinite-dimensional
vector space over $\mathbb{Q}$. 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 $\mathrm{Gal}(K/k)$ and set up the
Galois correspondence using *closed* subgroups of $\mathrm{Gal}(K/k)$. Using this
trick, every algebraic number field corresponds to some closed subgroup
of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.

So, for people studying algebraic number fields,
$$\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$$
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 $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ 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
$\mathrm{Gal}(K/k)$ 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 $\mathbb{Q}$ is contained in a cyclotomic field. So, they all sit
inside a field called $\mathbb{Q}^{\mathrm{cyc}}$, which is gotten by taking the rationals
and throwing in *all* $n$th roots of unity for *all* $n$. Since
$$\mathrm{Gal}(\mathbb{Q}(1^{\frac1n})/\mathbb{Q}) = (\mathbb{Z}/n)^\times$$
we know from Galois theory that $\mathrm{Gal}(\mathbb{Q}^{\mathrm{cyc}}/\mathbb{Q})$ must be a big group
containing all the groups $(\mathbb{Z}/n)^\times$ as closed subgroups. It's easy to see
that $(\mathbb{Z}/n)^\times$ is a quotient group of $(\mathbb{Z}/m)^\times$ if $m$ is divisible by $n$; this
lets us take the "inverse limit" of all the groups $(\mathbb{Z}/m)^\times$ --- and
that's $\mathrm{Gal}(\mathbb{Q}^{\mathrm{cyc}}/\mathbb{Q})$. This inverse limit is also the multiplicative
group of the ring $\widehat{\mathbb{Z}}$, the inverse limit of all the rings $\mathbb{Z}/n$. $\widehat{\mathbb{Z}}$
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:
$$\mathrm{Gal}(\mathbb{Q}^{\mathrm{cyc}}/\mathbb{Q}) = \widehat{\mathbb{Z}}^\times$$
and if we stay inside $\mathbb{Q}^{\mathrm{cyc}}$, 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 $\mathbb{Q}^{\mathrm{cyc}}$ is to $\overline{\mathbb{Q}}$ as homology theory is to
homotopy theory: it's all about *abelian* Galois groups, so it's
manageable.

People now use $\mathbb{Q}^{\mathrm{cyc}}$ as a kind of base camp for further expeditions
into the depths of $\overline{\mathbb{Q}}$. In particular, since
$$\mathbb{Q} \subset \mathbb{Q}^{\mathrm{cyc}} \subset \overline{\mathbb{Q}}$$
we get an exact sequence of Galois groups:
$$1 \to \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}^{\mathrm{cyc}}) \to \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{Gal}(\mathbb{Q}^{\mathrm{cyc}}/\mathbb{Q}) \to 1$$
So, to understand $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ we need to understand $\mathrm{Gal}(\mathbb{Q}^{\mathrm{cyc}}/\mathbb{Q})$,
$\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}^{\mathrm{cyc}})$ and how they fit together! The last two steps are not
so easy. Shafarevich has conjectured that $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}^{\mathrm{cyc}})$ is the
profinite completion of a free group, say $\widehat{F}$. This would give
$$1 \to \widehat{F} \to \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathbb{Z}^\times \to 1$$
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 $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, 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 $\mathbb{Q}$_, 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/](http://www.ams.org/joursearch/servlet/DoSearch?sendit=Search&f1=msc&v1=&co1=and&f2=title&v2=&co2=and&f3=anywhere&v3=&co3=and&f4=author&v4=cartier&jrnl=all&onejrnl=&startmo=00&startyr=0000&endmo=00&endyr=0000&format=standard&sperpage=30&cperpage=50&ssort=d&cso)

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`](http://www.arXiv.org/abs/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 $\mathbb{F}$ 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 $\mathbb{Q}(2^{1/3})$
has dimension 3 as a vector space over $\mathbb{Q}$, while quadratic extensions
have dimension $2^n$. 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.

------------------------------------------------------------------------

> *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