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.

In a previous paper, we have given asymptotic formulas for the number of isomorphism classes of $D_4$-extensions with discriminant up to a given bound, both when the signature of the extensions is or is not specified. We have also given very efficient exact formulas for this number when the signature is not specified. The aim of this paper is to give such exact formulas when the signature is specified. The problem is complicated by the fact that the ray class characters which appear are not all genus characters....

We give an explicit construction of an integral basis for a radical function field $K=k(t,\rho )$, where ${\rho }^n=D\in k\left[t\right]$, under the assumptions $[K:k(t\left)\right]=n$ and $char\left(k\right)\nmid n$. The field discriminant of $K$ is also computed. We explain why these questions are substantially easier than the corresponding ones in number fields. Some formulae for the $P$-signatures of a radical function field are also discussed in this paper.

We prove that van Hoeij’s original algorithm to factor univariate polynomials over the rationals runs in polynomial time, as well as natural variants. In particular, our approach also yields polynomial time complexity results for bivariate polynomials over a finite field.

Let $K$ be a number field containing, for some prime $\ell$, the $\ell$-th roots of unity. Let $L$ be a Kummer extension of degree $\ell$ of $K$ characterized by its modulus $𝔪$and its norm group. Let $K_{𝔪}$ be the compositum of degree $\ell$ extensions of $K$ of conductor dividing $𝔪$. Using the vector-space structure of $Gal(K_{𝔪}/K)$, we suggest a modification of the rnfkummer function of PARI/GP which brings the complexity of the computation of an equation of $L$ over $K$ from exponential to linear.

Let $𝒪$ be the maximal order of the cubic field generated by a zero $\epsilon$ of $x^3+(\ell -1)x^2-\ell x-1$ for $\ell \in \mathbb{Z}$, $\ell \ge 3$. We prove that $\epsilon ,\epsilon -1$ is a fundamental pair of units for $𝒪$, if $[𝒪:\mathbb{Z}[\epsilon \left]\right]\le \ell /3.$

We study infinite translation surfaces which are $\mathbb{Z}$-covers of compact translation surfaces. We obtain conditions ensuring that such surfaces have Veech groups which are Fuchsian of the first kind and give a necessary and sufficient condition for recurrence of their straight-line flows. Extending results of Hubert and Schmithüsen, we provide examples of infinite non-arithmetic lattice surfaces, as well as surfaces with infinitely generated Veech groups.

While most algebra is done by writing text and formulas, diagrams have always been used to present structural information clearly and concisely. Text shells are the de facto interface for computational algebraic number theory, but they are incapable of presenting structural information graphically. We present GiANT, a newly developed graphical interface for working with number fields. GiANT offers interactive diagrams, drag-and-drop functionality, and typeset formulas.

We present an algorithm for computing discriminants and prime ideal decomposition in number fields. The algorithm is a refinement of a $p$-adic factorization method based on Newton polygons of higher order. The running-time and memory requirements of the algorithm appear to be very good.

Let $\mathbb{Q}\left(\alpha \right)$ and $\mathbb{Q}\left(\beta \right)$ be algebraic number fields. We describe a new method to find (if they exist) all isomorphisms, $\mathbb{Q}\left(\beta \right)\to \mathbb{Q}\left(\alpha \right)$. The algorithm is particularly efficient if there is only one isomorphism.