Let $k$ be a number field, ${𝒪}_k$ its ring of integers, and $f(t,X)\in k\left(t\right)\left[X\right]$ be an irreducible polynomial. Hilbert’s irreducibility theorem gives infinitely many integral specializations $t\mapsto \overline{t}\in {𝒪}_k$ such that $f(\overline{t},X)$ is still irreducible. In this paper we study the set ${\mathrm{Red}}_f\left({𝒪}_k\right)$ of those $\overline{t}\in {𝒪}_k$ with $f(\overline{t},X)$ reducible. We show that ${\mathrm{Red}}_f\left({𝒪}_k\right)$ is a finite set under rather weak assumptions. In particular, previous results obtained by diophantine approximation techniques, appear as special cases of some of our results. Our method is different. We use elementary group...

The paper presents a careful analysis of the Cantor-Zassenhaus polynomial factorization algorithm, thus obtaining tight bounds on the performances, and proposing useful improvements. In particular, a new simplified version of this algorithm is described, which entails a lower computational cost. The key point is to use linear test polynomials, which not only reduce the computational burden, but can also provide good estimates and deterministic bounds of the number of operations needed for factoring....

Elliptic curves with CM unveil a new phenomenon in the theory of large algebraic fields. Rather than drawing a line between $0$ and $1$ or $1$ and $2$ they give an example where the line goes beween $2$ and $3$ and another one where the line goes between $3$ and $4$.

A variety $X$ over a field $K$ is of Hilbert type if $X\left(K\right)$ is not thin. We prove that if $f:X\to S$ is a dominant morphism of $K$-varieties and both $S$ and all fibers $f^{-1}\left(s\right)$, $s\in S\left(K\right)$, are of Hilbert type, then so is $X$. We apply this to answer a question of Serre on products of varieties and to generalize a result of Colliot-Thélène and Sansuc on algebraic groups.

We prove that if $K$ is a number field and $N$ is a Galois extension of $\mathbb{Q}$ which is not algebraically closed, then $N$ is not PAC over $K$.

We show explicit forms of the Bertini-Noether reduction theorem and of the Hilbert irreducibility theorem. Our approach recasts in a polynomial context the geometric Grothendieck good reduction criterion and the congruence approach to HIT for covers of the line. A notion of “bad primes” of a polynomial P ∈ ℚ[T,Y] irreducible over ℚ̅ is introduced, which plays a central and unifying role. For such a polynomial P, we deduce a new bound for the least integer t₀ ≥ 0 such that P(t₀,Y) is irreducible...

Dans cet article, nous tentons de généraliser à d’autres situations l’isomorphisme de groupes topologiques qui existe entre le groupe ${\displaystyle ℝ/\mathbb{Z}}$ et le groupe unitaire $𝕌=\{z\in ℂ/|z|=1\}$.Nous montrons que cet isomorphisme existe algébriquement en toute généralité : pour tout corps algébriquement clos $C$ et toute involution $c$ de $C$ les groupes $𝕌(C,c)=\{z\in C/zc(z)=1\}$ et ${\displaystyle C^{\<c\>}/\mathbb{Z}}$ sont isomorphes. Nous donnons ensuite un exemple d’involution $c_0$ de $ℂ$ qui n’est pas conjuguée, dans le groupe $\text{Aut}\left(ℂ\right)$, à la conjugaison complexe et telle que $𝕌(ℂ,c_0)$ soit topologiquement isomorphe...

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 $K$ over which the regular inverse Galois problem can be shown to be solvable, but such that $K$ does not contain a large field. The paper is complemented by model-theoretic observations on the diophantine nature of the regular inverse Galois problem.