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Let's start for a simple quote from wikipedia:

"No direct description is known for the absolute Galois group of the rational numbers. In this case, it follows from Belyi's theorem that the absolute Galois group has a faithful action on the dessins d'enfants of Grothendieck (maps on surfaces), enabling us to "see" the Galois theory of algebraic number fields."

What does wikipedia mean exactly by "a direct description" of $\operatorname{Gal}\left(\overline{\mathbb Q}/\mathbb Q\right)$?

It seems that the absolute group of rationals is very important in mathematics, in fact there are several tools from algebraic geometry whereby we try to study it (I'm measuring the importance of an object simply estimating the amount of forces invested on investigations about its nature). But why to give this importance to this particular object? For example why $\operatorname{Gal}\left({\overline K}/K\right)$, where $K$ is a generic field, is not "beautiful" as $\operatorname{Gal}\left(\overline{\mathbb Q}/\mathbb Q\right)$?

Thanks in advance.

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  • $\begingroup$ What do you mean by 'generic field'? Do you mean studying $\text{Gal}(\bar K / K)$ without knowledge of $K$? You can't expect many interesting properties there. Or do you have specific $K$ in mind? $\endgroup$ – Magdiragdag May 22 '14 at 8:12
  • $\begingroup$ I don't have a specific $K$ in mind, maybe my sentence is a bit strange. But why the absolute group of $\mathbb Q$ is so studied? $\endgroup$ – Dubious May 22 '14 at 8:14
  • $\begingroup$ Your quote gives one reason one might be interested in this: "to 'see' the Galois theory of algebraic number fields". $\endgroup$ – Magdiragdag May 22 '14 at 8:16
  • $\begingroup$ Not only does it package together all finite extensions (so contains lots of arithmetic data about number fields by CFT) but number theorists are finding that representations of this group arise in many situations, for example in elliptic curves, modular forms etc. $\endgroup$ – fretty May 30 '14 at 10:09
  • $\begingroup$ Various forms of this question have been discussed quite a bit on MO, see e.g. here. If you look at some of the related questions on the side-bar, or look for questions with the tag absolute-galois-group, you'll find other interesting discussions. $\endgroup$ – tracing Jan 19 '15 at 13:56
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The field of rational numbers $\Bbb Q$ is usually the first field one encounters, and so extensions of $\Bbb Q$ like $\Bbb Q(i)$ or $\Bbb Q(\sqrt 2)$ are the first manifestations of Galois groups one encounters.

Even then, the next basic fields are the finite fields $\Bbb F_p$ (of which $Gal_{\Bbb F_p}(\overline{\Bbb F_p} )$ is well understood), and then local fields like $\Bbb Q_p$ (whose absolute Galois group is a bit more complicated, but still "simple").

While not much can happen in finite or local fields, the Galois theory of global fields ($\Bbb Q, \Bbb F_p(X), \Bbb C(X), \ldots$) is much richer.

When you being studying field extensions of $\Bbb Q$ without Galois theory, there are "hard" questions like "okay, if I adjoin a root of $-1$ or a root of $2$ I get a degree $2$ extension, but what about adding both at once ? How do I know that $\sqrt 2 \notin \Bbb Q(i)$ or vice versa ?", or even harder, "how can I know that $\sqrt {11} \notin \Bbb Q(\sqrt 2,\sqrt 3, \sqrt 5,\sqrt 7)$ ?" (you can find those kind of questions on this website).
The general answer to these questions rely on understanding the Galois group of $\Bbb Q(\sqrt{-1},\sqrt 2,\sqrt 3,\sqrt 5,\ldots)$, and you can translate this in terms of the absolute Galois group by giving an explicit description of $\hom (Gal_\Bbb Q(\overline {\Bbb Q}) , \Bbb Z/2\Bbb Z)$.

A very related kind of question is what happens locally in extensions of global fields, or "what primes factors in $\Bbb Q(\sqrt 7)$ ?" and more generally "how do primes of $K$ factor in primes of $L$ in an algebraic extension $K \subset L$ ?". For our quadratic extensions, this is summed up by Gauss' quadratic reciprocity law, (a result that seemed unbelievable when I first heard about it).

A theorem of Kronecker says that every abelian extension of $\Bbb Q$ is in a cyclotomic extension. If we understand cyclotomic extensions (and we do) then we understand morphisms from $G$ to finite abelian groups, which means we understand the abelianization of $G$. To put it concisely $\Bbb Q \subset \Bbb Q(\zeta_n)$ has Galois group $(\Bbb Z/n\Bbb Z)^*$, and we know how a prime $(p)$ behaves in the extension by looking at what $p$ (we pick the positive generator) modulo $n$ does in that group.

A big achievement of the 20th century was to generalize this result to any global field, notably number fields other than $\Bbb Q$, so we have a "nice" description of $G^{ab}$ and a reciprocity map $\{primes\}\to G^{ab}$.

With all of this we can answer new questions like "But how do I know that $\sqrt[3]{11} \notin \Bbb Q(\sqrt[3]2, \sqrt[3]3,\sqrt[3]5,\sqrt[3]7) ?"$ or "when is $2$ a cube modulo $p$ ?" because this is about abelian extensions of $\Bbb Q(\sqrt{-3})$

So far we understand pretty well the morphisms (representations) from the absolute Galois groups to finite abelian groups. The question about representations into more complicated groups is an active area of research and an answer about those would give us tools to answer easily more of those kind of "simple" questions.

For example if $f(a)$ is the positive root of $x^5-x-a$, I don't think we have an easy answer to "But how do I show that $f(5) \notin \Bbb Q(f(1),f(2),f(3),f(4))$ ?"

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  • $\begingroup$ Do you have a reference for learning about the structure of $Gal(\overline{Q}_p/\mathbb{Q})$? $\endgroup$ – 54321user May 28 '17 at 0:15
  • $\begingroup$ @MoarCake559 i guess you can start by reading what's over there math.stackexchange.com/questions/495779/… $\endgroup$ – mercio May 28 '17 at 20:47
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A direct description means some kind of presentation, where we can explicitly write down what the elements look like and how to compose them. For instance, for odd primes the absolute Galois group of $\Bbb Q_p$ can be given an explicit presentation with generators and relations. Or for instance the group ${\rm Gal}(\overline{\Bbb F_p}/\Bbb F_p)\cong\widehat{\Bbb Z}$ (canonically via Frobenius), the profinite integers, which by CRT can be decomposed as vectors from $\prod \Bbb Z_p$, and $p$-adic integers from $\Bbb Z_p$ have explicit $p$-adic expansions.

Since $\Bbb Z$ and $\Bbb Q$ are where all of number theory began, it makes sense to place special importance on the Galois group $G={\rm Gal}(\overline{\Bbb Q}/\Bbb Q)$, if only for sentimental reasons. It is essentially the full symmetry group of numbers themselves, where the structure preserved is the truth of polynomial equations.

Other fields have only shadows of this symmetry (algebraic number fields), are only "number-like" by virtue of having analogous properties (global function fields) or merely by virtue of being a field, or involve transcendentals which eschew polynomial equations (like $\Bbb C$ taken as a whole). While modern understanding and experience with the spread of number theory beyond just "numbers" informs us we may as well place just as much importance on other Galois groups and consider them just as beautiful, we have a special place in our hearts for numbers over their cousins.

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You may be interested in the theory of Dessin's d'Enfants which are a tool for studying $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.


From interview with Richard Taylor and Scott Sheffield.

HOW LONG HAVE YOU BEEN AT HARVARD?

Just over two weeks.

WHERE ARE YOU FROM?

I spent most of my career in Cambridge, but last year I moved to Oxford for one year before coming here. Before that I did a one year post-doc in Paris, a PhD in Princeton and was an undergraduate in Cambridge.

WHAT ARE YOUR MAJOR RESEARCH INTERESTS AND ACHIEVEMENTS?

The great problem that motivates me is to understand the absolute Galois group of the rational numbers, that is, the group of all automorphisms of the field of algebraic numbers (complex numbers which are the roots of nonzero polynomials with rational coefficients). If you like you can talk about all Galois groups of finite extensions of the rational numbers, but this is a convenient way to put them all together. It doesn't make a lot of difference, but it is technically neater to put them all together. The question that has motivated almost everything I have done is, "What's the structure of that group?" One of the great achievements of mathematicians of the first half of this century is called class field theory, and one way of seeing it is as a description of all abelian quotients of the absolute Galois group of Q, or if you like, the classification of the abelian extensions of the field of the rational numbers. That's only a very small part of this group. The group is extremely complicated, and just describing the abelian part doesn't solve the problem. For instance John Thompson proved that the monster group is a quotient group of this group in infinitely many ways.

We gather that many people view $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ as a generalization of Class Field Theory.

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  • $\begingroup$ This neither answers (or attempts to answer) the question (why we place relative importance on the Galois group of rationals over other fields), nor does it provide any new information to the OP or any other readers (since OP already mentions dessin's in the question). $\endgroup$ – blue May 27 '14 at 17:14
  • $\begingroup$ @GrigoryM His selection from Wikipedia doesn't talk about a direct description but it does talk about a faithful action which is a direct reference to dessins. In any case, my response doesn't explain why people study the absolute Galois group, just what tools they use. The existence of a tool for "seeing" absolute Galois theory makes it much more concrete than other problems maybe. $\endgroup$ – cactus314 May 27 '14 at 18:46

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