"Understanding" $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$

$\newcommand{\bb}{\mathbb}$ This is a response to Charles' remark to JSE's answer, why doesn't $\bar{\bb Q}$ come with a standard algebraic closure inside the complex numbers $\bb C$?

First, if one considers an abstract extension $K/\bb Q$, then $K$ has $d = [K:\bb Q]$ embeddings into the complex numbers, which can not, a priori, be distinguished in any way. (e.g. maps from $K = \bb Q[x]/(x^3-2)$ to $\bb C$ require a "choice" of $2^{1/3}$ in $\bb C$.

Of course, this doesn't answer Charles' question, which, I imagine, is more along the following lines. Why doesn't one simply start with the complex numbers, and then consider the set of algebraic numbers inside $\bb C$? The resulting field is clearly isomorphic to $\bar{\bb Q}$, and, moreover, comes with a canonical embedding into $\bb C$.

The problem arises when one wants to define Frobenius elements. Defining such elements amounts to giving a choice of embedding from $\bar{\bb Q}$ into $\bar{\bb Q}_p$. So there is a choice to be made for every $p$! Thinking of $\bar{\bb Q}$ inside $\bb C$ fixes this choice for "$p=0$" only.

To make this completely explicit, consider the splitting field $K$ of $x^3 - 2$. In the "fields live inside $\bb C$" optic, $K$ is the field $\bb Q(2^{1/3},\sqrt{-3})$ where $2^{1/3}$ is real and the imaginary part of $\sqrt{-3}$ is positive. Clearly $\mathrm{Gal}(K/\bb Q) = S_3$, where we can think of $(123)$ as sending $2^{1/3}\mapsto e^{2\pi i/3} 2^{1/3}$. If $p = 7$, then $\mathrm{Frob}_p$ has order $3$ in $S_3$. We can ask, does $\mathrm{Frob}_p = (123)$ or $(132)$? We find that $$ \mathrm{Frob}_p = (123) \quad\mbox{if } \sqrt{-3}\equiv 2\mod 7 $$ $$ \mathrm{Frob}_p = (132) \quad\mbox{if } \sqrt{-3}\equiv 5\mod 7 $$ and knowing that the imaginary part of $\sqrt{-3}$ is positive does not allow us to determine $\mathrm{Frob}_p$ without choosing an embedding of $K$ into $\bar{\bb Q}_7$.

Taylor said that he's never heard of anyone proposing a similar direct description of $G$ and that to understand $G$ one studies the representations of $G$.

I remember Mazur telling me this when I was a grad student. He made this point in the following way. You shouldn't really think of $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ as a group which has elements, but as a "group up to conjugacy" - thus, the aspects of Galois groups that really make sense to think about are the conjugacy-invariant things: conjugacy classes (like Frobenii) and representations.

To unpack this a bit more: a Galois group is a fundamental group. But to talk about a fundamental group (as opposed to a groupoid) you need to choose a basepoint. To talk about an absolute Galois group you also need to choose a basepoint, which is to say an algebraic closure $\bar{\mathbb{Q}}/\mathbb{Q}$. (So just as one should talk about $\pi_1(X,*)$ rather than $\pi_1(X)$, one should talk about $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ rather than $\mathrm{Gal}(\mathbb{Q})$.) But a basepoint you can just draw with a pencil. A Galois closure of $\mathbb{Q}$ is not so easy.

What would it mean to understand this Galois group? You could mean several things.

You could mean trying to give the group in terms of some smallish generators and relations. This would be nice, and help to answer questions like the inverse Galois problem that Greg Muller mentioned, and having a certain family of "generating" Galois automorphisms would allow you to study questions about e.g. the representation theory in quite explicit terms. However, the Galois group is an uncountable profinite group, and so to give any short description in terms of generators and relations leads you into subtle issues about which topology you want to impose.

You could also ask for a coherent system of names for all Galois automorphisms, so that you can distinguish them and talk about them on an individual basis. One system of names comes from the dessins d'enfant that Ilya mentioned: associated to a Galois automorphism we have some associated data.

• We have its image under the cyclotomic character, which tells us how it acts on roots of unity. By the Kronecker-Weber theorem this tells us about the abelianization of the Galois group.
• We also have an element in the free profinite group on two generators, which (roughly speaking) tells us something about how abysmally acting on the coefficients of a power series fails to commute with analytic continuation.

These two names satisfy some relations, called the $2$-, $3$-, and $5$-cycle relation, which are conjectured to generate all relations (at least the last time I checked), but it is difficult to know whether they actually do so. If they do, then the Galois group is the so-called Grothendieck-Teichmüller group.

The problem with this perspective is that the names aren't very explicit (and we don't expect them to be: we may need the axiom of choice to show they exist, and there are only two Galois automorphisms of $\mathbb{C}$ that are measurable functions!) and it seems to be a difficult problem to determine whether the Grothendieck-Teichmuller group really is the whole thing. (Or it was the last time I checked.)

However, the cyclotomic character is a nice, and fairly canonical, name associated for Galois automorphisms. We could try to generalize this: there are Kummer characters telling us what a Galois automorphism does to the system of real positive roots of a positive rational number number (these determine a compatible system of roots of unity, or equivalent an element of the Tate module of the roots of unity). This points out one of the main difficulties, though: we had to make choices of roots of unity to act on, and if Galois theory taught us nothing else it is that different choices of roots of an irreducible polynomial should be viewed as indistinguishable. Different choices differ by conjugation in the Galois group.

This brings us to the point JSE was making: if we take the "symmetry" point of view seriously, we should only be interested in conjugacy-invariant information about the Galois group. Assigning names to elements or giving a presentation doesn't really mesh with the core philosophy.

So this brings us to how many people here have mentioned understanding the Galois group: you understand it by how it manifests, in terms of its representations (as permutations, or on dessins, or by representations, or by its cohomology), because this is how it's most useful. Then you can study arithmetic problems by applying knowledge about this. If I have two genus $0$ curves over $\mathbb{Q}$, what information distinguishes them? If I have two lifts of the same complex elliptic curve to $\mathbb{Q}$, are they the same? How can I get information about a reduction of an abelian variety mod $p$ in terms of the Galois action on its torsion points? Et cetera.