Related papers: Note sur les corps 2-rationnels

We generalize the notion of a projective profinite group to a projective pair of a profinite group and a closed subgroup. We establish the connection with Pseudo Algebraically Closed (PAC) extensions of PAC fields: Let M be an algebraic…

We compute the first explicit polynomials with Galois groups $G=P\Gamma L_3(4)$, $PGL_3(4)$, $PSL_3(4)$ and $PSL_5(2)$ over $\mathbb{Q}(t)$. Furthermore we compute the first examples of totally real polynomials with Galois groups…

We consider function fields of transcendence degree at least 2 over algebraic closures of finite fields, and describe a functorial way to recover such function fields form their pro-l Galois theory.

Let F/k be a Galois extension of number fields with dihedral Galois group of order 2q, where q is an odd integer. We express a certain quotient of S-class numbers of intermediate fields, arising from Brauer-Kuroda relations, as a unit…

We observe that some basic but fundamental constructions in Galois theory can be used to obtain some interesting restrictions on the structure of Galois groups of maximal $p$-extensions of fields containing a primitive $p$th root of unity.…

We associate to every algebraic number field a hyperbolic surface lamination and an external fundamental group: the latter a generalization of the fundamental germ that necessarily contains external (not first order definable) elements. The…

Given a natural number n and a number field K, we show the existence of an integer \ell_0 such that for any prime number \ell\geq \ell_0, there exists a finite extension F/K, unramified in all places above \ell, together with a principally…

We compute the Galois group of a polynomial whose roots are determined by the critical points of a scalar potential in type IIB compactifications. We focus our study on certain perturbative models where it is feasible to construct a de…

We show that finite Galois extensions with cyclic Galois group are radical.

In this note we improve the theorem on Galois rational covers $X\dashrightarrow V$ for primitive Fano varieties $V$, recently proven by the author, in the two directions: we extend to the maximum the class of Galois groups $G$, for which…

Let G be the absolute Galois group of a global field. Let r1 and r2 be two p-adic, finite dimensional representations of G. Then there exists a finite number of primes q such that if the characteristic polynomials of r1(Frob_q) and…

In this paper, we develop an explicit method to express finite algebraic numbers (in particular, certain idempotents among them) in terms of linear recurrent sequences, and give applications to the characterization of the splitting primes…

We determine the Galois module structure of the parameterizing space of elementary $p$-abelian extensions of a field $K$ when $\text{Gal}(K/F)$ is any finite $p$-group, under the assumption that the maximal pro-$p$ quotient of the absolute…

Let $G$ be a finite group and let $ram^{t}(G)$ denote the minimal positive integer $n$ such that $G$ can be realized as the Galois group of a tamely ramified extension of $\mathbb{Q}$ ramified only at $n$ finite primes. Let $d(G)$ denote…

For many finite groups, the Inverse Galois Problem can be approached through modular/automorphic Galois representations. This is a report explaining the basic strategy, ideas and methods behind some recent results. It focusses mostly on the…

We consider the distribution of the Galois groups $\operatorname{Gal}(K^{\operatorname{un}}/K)$ of maximal unramified extensions as $K$ ranges over $\Gamma$-extensions of $\mathbb{Q}$ or $\mathbb{F}_q(t)$. We prove two properties of…

Let p be an odd prime satisfying Vandiver's conjecture. We consider two objects, the Galois group X of the maximal unramified abelian pro-p extension of the compositum of all Z_p-extensions of the pth cyclotomic field and the Galois group G…

We compute the semisimplifications of the mod-$p$ reductions of $2$-dimensional crystalline representations of the absolute Galois group of the p-adic numbers of slope $(2,3)$ and arbitrary weight, building on work of Bhattacharya-Ghate

We consider lifting of mod p representations to mod p^2 representations in the setting of representations of (i) finite groups; (ii) absolute Galois groups of abstract fields; and (iii) absolute Galois groups of local and global fields.

The aim of this paper is to extend our old results about Galois action on the torsion points of abelian varieties to the case of (finitely generated) fields of characteristic 2.