TITLE: Why there is much interest in the study of $\operatorname{Gal}\left(\overline{\mathbb Q}/\mathbb Q\right)$?
QUESTION [37 upvotes]: 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.

REPLY [9 votes]: 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.