In the previous paper [Sch2] it has been shown that ray class fields over quadratic imaginary number fields can be generated by simple products of singular values of the Klein form defined below. In the present article the second named author has constructed more general products that are contained in ray class fields thereby correcting Theorem 2 of [Sch2]. An algorithm for the computation of the algebraic equations of the numbers in Theorem 1 of this paper has been implemented in a KASH program...

We determine the structures of the Galois groups Gal$(K_{ur}/K)$ of the maximal unramified extensions $K_{ur}$ of imaginary quadratic number fields $K$ of conductors $\leqq 420(\leqq 719$ under the Generalized Riemann Hypothesis). For all such $K$, $K_{ur}$ is $K$, the Hilbert class field of $K$, the second Hilbert class field of $K$, or the third Hilbert class field of $K$. The use of Odlyzko’s discriminant bounds and information on the structure of class groups obtained by using the action of Galois groups on class groups is essential. We also use class...

Le but de cet article est l’étude des corps cycliques quintiques définis par les polynômes d’E. Lehmer. On calcule premièrement le conducteur de ces corps dans le cas général (non nécessairement premier) puis on généralise un théorème (qui donne les unités de ces corps) démontré par R. Schoof et L.C. Washington. Par la méthode de dévissage des unités cyclotomiques, qui calcule le nombre de classes et les unités, on dresse une table de ces corps particuliers (de conducteur $f\le 3000000$) et de leur nombre de...

Assuming the Generalized Riemann Hypothesis (GRH), we show that the norm-Euclidean Galois cubic fields are exactly those with discriminant$$\Delta =7^2,9^2,{13}^2,{19}^2,{31}^2,{37}^2,{43}^2,{61}^2,{67}^2,{103}^2,{109}^2,{127}^2,{157}^2.$$A large part of the proof is in establishing the following more general result: Let $K$ be a Galois number field of odd prime degree $\ell$ and conductor $f$. Assume the GRH for ${\zeta }_K\left(s\right)$. If$$38{(\ell -1)}^2{(logf)}^{6}loglogf\<f,$$then $K$ is not norm-Euclidean.

We survey methods to compute three-point branched covers of the projective line, also known as Belyĭ maps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic methods, modular forms methods, and $p$-adic methods. Along the way, we pose several questions and provide numerous examples.

Let $\mathbb{Q}\left(\alpha \right)$ be an algebraic number field given by the minimal polynomial $f$ of $\alpha$. We want to determine all subfields $\mathbb{Q}\left(\beta \right)\subset \mathbb{Q}\left(\alpha \right)$ of given degree. It is convenient to describe each subfield by a pair $(g,h)\in \mathbb{Z}\left[t\right]\times \mathbb{Q}\left[t\right]$ such that $g$ is the minimal polynomial of $\beta =h\left(\alpha \right)$. There is a bijection between the block systems of the Galois group of $f$ and the subfields of $\mathbb{Q}\left(\alpha \right)$. These block systems are computed using cyclic subgroups of the Galois group which we get from the Dedekind criterion. When a block system is known we compute the corresponding...

We generalize the concept of reduced Arakelov divisors and define C-reduced divisors for a given number C ≥ 1. These C-reduced divisors have remarkable properties, similar to the properties of reduced ones. We describe an algorithm to test whether an Arakelov divisor of a real quadratic field F is C-reduced in time polynomial in $log|{\Delta }_F|$ with ${\Delta }_F$ the discriminant of F. Moreover, we give an example of a cubic field for which our algorithm does not work.

The class numbers h⁺ of the real cyclotomic fields are very hard to compute. Methods based on discriminant bounds become useless as the conductor of the field grows, and methods employing Leopoldt's decomposition of the class number become hard to use when the field extension is not cyclic of prime power. This is why other methods have been developed, which approach the problem from different angles. In this paper we extend one of these methods that was designed for real cyclotomic fields of prime...

We present algorithms for the computation of extreme binary Humbert forms in real quadratic number fields. With these algorithms we are able to compute extreme Humbert forms for the number fields $\mathbb{Q}\left(\sqrt{13}\right)$ and $\mathbb{Q}\left(\sqrt{17}\right)$. Finally we compute the Hermite-Humbert constant for the number field $\mathbb{Q}\left(\sqrt{13}\right)$.

We describe an algorithm due to Gauss, Shanks and Lagarias that, given a non-square integer $D\equiv 0,1$ mod $4$ and the factorization of $D$, computes the structure of the $2$-Sylow subgroup of the class group of the quadratic order of discriminant $D$ in random polynomial time in $log\left|D\right|$.