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...

Nous démontrons que dans la catégorie $ℱ$ des foncteurs entre espaces vectoriels sur ${𝔽}_2$, le produit tensoriel entre le second foncteur injectif standard non constant $V\mapsto {𝔽}_2^{{\left(V^{*}\right)}^{\oplus 2}}$ et un foncteur puissance extérieure est artinien. Seul était antérieurement connu le caractère artinien de cet injectif ; notre résultat constitue une étape pour l’étude du troisième foncteur injectif standard non constant de $ℱ$.Nous utilisons le foncteur de division par le foncteur identité et des considérations issues de la théorie...

We show that if G is a non-archimedean, Roelcke precompact Polish group, then G has Kazhdan's property (T). Moreover, if G has a smallest open subgroup of finite index, then G has a finite Kazhdan set. Examples of such G include automorphism groups of countable ω-categorical structures, that is, the closed, oligomorphic permutation groups on a countable set. The proof uses work of the second author on the unitary representations of such groups, together with a separation result for infinite permutation...

We consider the full group [φ] and topological full group [[φ]] of a Cantor minimal system (X,φ). We prove that the commutator subgroups D([φ]) and D([[φ]]) are simple and show that the groups D([φ]) and D([[φ]]) completely determine the class of orbit equivalence and flip conjugacy of φ, respectively. These results improve the classification found in [GPS]. As a corollary of the technique used, we establish the fact that φ can be written as a product of three involutions from [φ].