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A Theorem of Hou, Leung and Xiang generalised Kneser's addition Theorem to field extensions. This theorem was known to be valid only in separable extensions, and it was a conjecture of Hou that it should be valid for all extensions. We give an alternative proof of the theorem that also holds in the non-separable case, thus solving Hou's conjecture. This result is a consequence of a strengthening of Hou et al.'s theorem that is a transposition to extension fields of an addition theorem of Balandraud., 17 pages

Let X be an algebraic curve over Q and t a non-constant Q-rational function on X such that Q(t) is a proper subfield of Q(X). For every integer n pick a point P_n on X such that t(P_n)=n. We conjecture that, for large N, among the number fields Q(P_1), ..., Q(P_N) there are at least cN distinct. We prove this conjecture in the special case when t defines a geometrically abelian covering of the projective line, and the critical values of t are all rational. This implies, in particular, that our conjecture follows from a famous conjecture of Schinzel., Comment: Some typos are corrected. The article is now accepted in Periodica Math. Hungarica

International audience; We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in $\Ba \in L[y]$, where $L=\QQ(\alpha_1, \dots, \alpha_{\ell})$ is an algebraic extension of the rational numbers. Our bounds are single exponential in $\ell$ and match the ones presented in \cite{st-issac-2011} for the case $\ell=1$. We consider two approaches. The first, indirect approach, using multivariate resultants, computes a univariate polynomial with integer coefficients, among the real roots of which are the real roots of $\Ba$. The Boolean complexity of this approach is $\sOB(N^{4\ell+4})$, where $N$ is the maximum of the degrees and the coefficient bitsize of the involved polynomials. The second, direct approach, tries to solve the polynomial directly, without reducing the problem to a univariate one. We present an algorithm that generalizes Sturm algorithm from the univariate case, and modified versions of well known solvers that are either numerical or based on Descartes' rule of sign. We achieve a Boolean complexity of $\sOB(\min\set{N^{4\ell + 7},N^{2\ell^2+6}})$ and $\sOB( N^{2\ell+4})$, respectively. %% We implemented the algorithms in \func{C} as part of the core library of \mathematica and we illustrate their efficiency over various data sets.

Let K \subset L be a field extension. Given K-subspaces A,B of L, we study the subspace spanned by the product set AB = {ab | a \in A, b \in B}. We obtain some lower bounds on the dimension of this subspace and on dim B^n in terms of dim A, dim B and n. This is achieved by establishing linear versions of constructions and results in additive number theory mainly due to Kemperman and Olson., Comment: To appear in Linear and Multilinear Algebra

Contribution à un ouvrage.; A formula is given for solving by radicals any polynomial of degree 5 which is solvable by radicals. This formula is valid on any field of characteristic different from 2 and 5. The field extension which is generated by the radicals which appear in the result is always minimal, when only one root is produced, as well as when all roots are given. This formula has been implemented in Maple.