Tamely ramified extensions's structure.
The absolute Galois group of a pseudo p-adically closed field.
The absolute Galois group of a pseudo real closed field
The arithmetic of curves defined by iteration
We show how the size of the Galois groups of iterates of a quadratic polynomial f can be parametrized by certain rational points on the curves Cₙ: y² = fⁿ(x) and their quadratic twists (here fⁿ denotes the nth iterate of f). To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality theorem...
The distributivity property of finite intersections of valuation rings
The effect on associated prime ideals produced by an extension of the base field.
The field of Nash functions and factorization of polynomials
The algebraically closed field of Nash functions is introduced. It is shown that this field is an algebraic closure of the field of rational functions in several variables. We give conditions for the irreducibility of polynomials with Nash coefficients, a description of factors of a polynomial over the field of Nash functions and a theorem on continuity of factors.
The fundamental group of a Galois cover of .
The Galois Group of a Radical Extension of the Rationals.
The Galois group of generic symmetric matrices
The Galois relation x₁ = x₂ + x₃ for finite simple groups
The Galois relation x₁ = x₂+x₃ and Fermat over finite fields
The group of automorphisms of algebraic function fields.
The Hasse conjecture for cyclic extensions.
The inverse Galois problem and rational points on moduli spaces.
The local lifting problem for actions of finite groups on curves
Let be an algebraically closed field of characteristic . We study obstructions to lifting to characteristic the faithful continuous action of a finite group on . To each such a theorem of Katz and Gabber associates an action of on a smooth projective curve over . We say that the KGB obstruction of vanishes if acts on a smooth projective curve in characteristic in such a way that and have the same genus for all subgroups . We determine for which the KGB obstruction...
The Pythagorean closure of fields.
The regular inverse Galois problem over non-large fields
By a celebrated theorem of Harbater and Pop, the regular inverse Galois problem is solvable over any field containing a large field. Using this and the Mordell conjecture for function fields, we construct the first example of a field over which the regular inverse Galois problem can be shown to be solvable, but such that does not contain a large field. The paper is complemented by model-theoretic observations on the diophantine nature of the regular inverse Galois problem.
The Relative Separable Closure of a Valued Field in its Completion.
The Sylow p-Subgroups of Tame Kernels in Dihedral Extensions of Number Fields
Let F/E be a Galois extension of number fields with Galois group . In this paper, we give some expressions for the order of the Sylow p-subgroups of tame kernels of F and some of its subfields containing E, where p is an odd prime. As applications, we give some results about the order of the Sylow p-subgroups when F/E is a Galois extension of number fields with Galois group .