Cet article a pour objectif de présenter un algorithme permettant de montrer, à l’aide d’un ordinateur, l’euclidianité pour la norme du sous-corps réel maximal $K$ du corps cyclotomique $\mathbb{Q}\left({\zeta }_{32}\right)$ où ${\zeta }_{32}=e^{i\pi /16}$, corps totalement réel de degré $8$ et de discriminant $2147483648$, et plus précisément de prouver que $M\left(K\right)=\frac{1}{2}$. La méthode utilisée permet par ailleurs de prouver que pour $K=\mathbb{Q}({\zeta }_{16}+{\zeta }_{16}^{-1})$, on a également $M\left(K\right)=\frac{1}{2}$ (conjecture de H. Cohn et J. Deutsch). Les résultats relatifs à ce cas sont exposés en fin d’article.

Soit $A$ un anneau de Dedekind, de corps des fractions $K$, et soit $L$ une extension galoisienne de $K$, dont le groupe de Galois $G$ est cyclique d’ordre premier. On note $B$ la clôture intégrale de $A$ dans $L$. Il existe une unique décomposition du $A\left[G\right]$-module $B$ en somme directe de sous-modules indécomposables. On détermine cette décomposition lorsque $K$ est un corps local ou un corps de nombres. Le résultat dépend d’une part des caractères irréductibles de $G$ sur $K$, d’autre part des nombres de ramification associés...

We develop a new combinatorial method to deal with a degree estimate for subalgebras generated by two elements in different environments. We obtain a lower bound for the degree of the elements in two-generated subalgebras of a free associative algebra over a field of zero characteristic. We also reproduce a somewhat refined degree estimate of Shestakov and Umirbaev for the polynomial algebra, which plays an essential role in the recent celebrated solution of the Nagata conjecture and the strong...

Let ${\Delta }_{n,d}$ (resp. ${\Delta }_{n,d}^{\text{'}}$) be the simplicial complex and the facet ideal $I_{n,d}=(x_1\cdots x_d,x_{d-k+1}\cdots x_{2d-k},...,x_{n-d+1}\cdots x_n)$ (resp. $J_{n,d}=(x_1\cdots x_d,x_{d-k+1}\cdots x_{2d-k},...,x_{n-2d+2k+1}\cdots x_{n-d+2k},x_{n-d+k+1}\cdots x_n{x}_1\cdots x_k)$). When $d\ge 2k+1$, we give the exact formulas to compute the depth and Stanley depth of quotient rings $S/J_{n,d}$ and $S/I_{n,d}^t$ for all $t\ge 1$. When $d=2k$, we compute the depth and Stanley depth of quotient rings $S/J_{n,d}$ and $S/I_{n,d}$, and give lower bounds for the depth and Stanley depth of quotient rings $S/I_{n,d}^t$ for all $t\ge 1$.

This note summarizes a presentation made at the Third International Meeting on Integer Valued Polynomials and Problems in Commutative Algebra. All the work behind it is joint with Scott T. Chapman, and will appear in [2]. Let $\mathrm{Int}\left(\mathbb{Z}\right)$ represent the ring of polynomials with rational coefficients which are integer-valued at integers. We determine criteria for two such polynomials to have the same image set on $\mathbb{Z}$.

We prove a quantitative version of a result of Furstenberg [20] and Deligne [14] stating that the diagonal of a multivariate algebraic power series with coefficients in a field of positive characteristic is algebraic. As a consequence, we obtain that for every prime $p$ the reduction modulo $p$ of the diagonal of a multivariate algebraic power series $f$ with integer coefficients is an algebraic power series of degree at most $p^A$ and height at most $A{p}^A$, where $A$ is an effective constant that only depends on...

Let $H$ be a Krull monoid with finite class group where every class contains some prime divisor. It is known that every set of lengths is an almost arithmetical multiprogression. We investigate which integers occur as differences of these progressions. In particular, we obtain upper bounds for the size of these differences. Then, we apply these results to show that, apart from one known exception, two elementary $p$-groups have the same system of sets of lengths if and only if they are isomorphic.