From its early beginnings up to nowadays, algebraic number theory has evolved in symbiosis with Galois theory: indeed, one could hold that it consists in the very study of the absolute Galois group of the field of rational numbers. Nothing like that can be said of transcendental number theory. Nevertheless, couldn’t one associate conjugates and a Galois group to transcendental numbers such as π...

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 index. Our formula is valid for arbitrary extensions with Galois group D2q and for arbitrary Galois-stable sets of primes S , containing the Archimedean ones. Ou...

We give the first polynomial-time algorithm for checking whether the Galois group Gal (f) of an input polynomial f(X) ∈ Q[X] is nilpotent: the running time of our algorithm is bounded by a polynomial in the size of the coefficients of f and the degree of f . Additionally, we give a deterministic polynomial-time algorithm that, when given as input a polynomial f(X) ∈ Q[X] with nilpotent Galois g...

We study Galois points for a hypersurface X with dimSing(X) ≤ dimX − 2. The purpose of this article is to determine the set ∆(X) of Galois points in characteristic zero: Indeed, we give a sharp upper bound of the number of Galois points in terms of dimX and dimSing(X) if ∆(X) is a finite set, and prove that X is a cone if ∆(X) is infinite. To achieve our purpose, we need a certain hyperplane se...

For each transitive permutation group G on n letters with n ≤ 4, we give without proof results, conjectures, and numerical computations on discriminants of number fields L of degree n over Q such that the Galois group of the Galois closure of L is isomorphic to G.

By two well-known results, one of Ax, one of Lubotzky and van den Dries, a profinite group is projective iff it is isomorphic to the absolute Galois group of a pseudo-algebraically closed field. This paper gives an analogous characterization of relatively projective profinite groups as absolute Galois groups of regularly closed fields.

(a) Find Galois points and the Galois groups for singular plane curves. – for smooth curves, the number of Galois points is at most three (resp. four) if they are outer (resp. inner). The Galois groups are cyclic. [46, 62] – (i) How is the structure of Galois group and how many Galois points do there exist? Is it true that the maximal number of outer (resp. inner) Galois points is three (resp. ...