Profinite groups as absolute Galois groups

This is a very good question which is a big open problem. There are a number of theorems, some of them easy and some very difficult, and also a number of conjectures, restricting the class of groups which may turn out to be absolute Galois groups. But it seems that nobody has any idea about how a precise description of the class of absolute Galois groups might look like.

One general point-of-view position on which the experts seem to agree is that the right object to deal with is not just the Galois group $G_K=\operatorname{Gal}(\overline{K}|K)$ considered as an abstract profinite group, but the pair $(G_K,\chi_K)$, where $\chi_K\colon G_K\to \widehat{\mathbb Z}^*$ is the cyclotomic character of the group $G_K$ (describing its action in the group of roots of unity in $\overline K$). The question about being absolute Galois should be properly asked about pairs $(G,\chi)$, where $G$ is a profinite group and $\chi\colon G\to \widehat{\mathbb Z}^*$ is a continuous homomorphism, rather than just about the groups $G$.

For example, here is an important and difficult theorem restricting the class of absolute Galois groups: for any field $K$, the cohomology algebra $H^*(G_K,\mathbb F_2)$ is a quadratic algebra over the field $\mathbb F_2$. This is (one of the formulations of) the Milnor conjecture, proven by Rost and Voevodsky. More generally, for any field $K$ and integer $m\ge 2$, the cohomology algebra $\bigoplus_n H^n(G_K,\mu_m^{\otimes n})$ is quadratic, too, where $\mu_m$ denotes the group of $m$-roots of unity in $\overline K$ (so $G_K$ acts in $\mu_m^{\otimes n}$ by the character $\chi_K^n$). This is the Bloch-Kato conjecture, also proven by Rost and Voevodsky.

Here is a quite elementary general theorem restricting the class of absolute Galois groups: for any field $K$ of at most countable transcendence degree over $\mathbb Q$ or $\mathbb F_p$, the group $G_K$ has a decreasing filtration $G_K\supset G_K^0\supset G_K^1\supset G_K^2\supset\dotsb$ by closed subgroups normal in $G_K$ such that $G_K=\varprojlim_n G_K/G_K^n$ and $G_K/G_K^0$ is either the trivial group or $C_2$, while $G_K^n/G_K^{n+1}$, $n\ge0$ are closed subgroups in free profinite groups. (Groups of the latter kind are called "projective profinite groups" or "profinite groups of cohomological dimension $\le1$".) One can get rid of the assumption of countability of the transcendence degree by considering filtrations indexed by well-ordered sets rather than just by the integers.

Here is a conjecture about Galois groups of arbitrary fields (called "the generalized/strengthened version of Bogomolov's freeness conjecture). For any field $K$, consider the field $L=K[\sqrt[\infty]K]$ obtained by adjoining to $K$ all the roots of all the polynomials $x^n-a$, where $n\ge2$ and $a\in K$. In particular, when the field $K$ contains all the roots of unity, the field $L$ is (the maximal purely inseparable extension of) the maximal abelian extension of $K$. Otherwise, you may want to call $L$ "the maximal radical extension of $K$". The conjecture claims that the absolute Galois group $G_L=\operatorname{Gal}(\overline{L}|L)$ is a projective profinite group.

In order to study absolute Galois groups of fields one may also use another consequence of the aforementioned Rost-Voevodsky theorem: in fact, for any field $K$, the cohomology algebra $H^*(G_K,\mathbb{F}_p)$ is a quadratic algebra over the field $\mathbb{F}_p$ also for $p$ odd; and this is still true for the algebra $H^*(G_K(p),\mathbb{F}_p)$, where $G_K(p)$ denotes the maximal pro-$p$ quotient of $G_K$, if $K$ contains a primitive $p$-th root of 1 - i.e., $G_K(p)$ is the Galois group of the compositum of all Galois $p$-extensions $L/K$, and it is called the maximal pro-$p$ Galois group of $K$. Note that the class of such Galois pro-$p$ groups includes all absolute Galois groups which are pro-$p$.

One reduces to pro-$p$ groups as in general they are easier to deal with than profinite groups.

Pro-$p$ groups whose $\mathbb{F}_p$-cohomology is a quadratic algebra - and thus they are «good candidates» for being realized as maximal pro-$p$ Galois groups - are studied by S. Chebolu, I. Efrat and J. Minàc in the paper Quotients of absolute Galois groups which determine the entire Galois cohomology (2012), and in my paper Bloch-Kato pro-$p$ groups and locally powerful groups (2014). Both papers are available also on the arXiv.

Also, recently I. Efrat, E. Matzri, J. Minàc and N.D. Tan proved that the $\mathbb{F}_p$-cohomology algebra of maximal pro-$p$ Galois groups of fields (containing a primitive $p$-th root of 1) satisfies another property, called the «vanishing of triple Massey products»: in the papers Triple Massey products and Galois theory and Triple Massey products vanish over all fields Minàc and Tan produce some examples of pro-$p$ groups which are not realizable as maximal pro-$p$ Galois groups (and thus as absolute Galois groups).

All the papers I mentioned are available in the arXiv: if you wish I may provide the links.