For any square-free positive integer $m\equiv 10(mod16)$ with $m\ge 26$, we prove that the class number of the real cyclotomic field $\mathbb{Q}({\zeta }_{4m}+{\zeta }_{4m}^{-1})$ is greater than $1$, where ${\zeta }_{4m}$ is a primitive $4m$th root of unity.

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 $log\left|D\right|$.

The study of class number invariants of absolute abelian fields, the investigation of congruences for special values of L-functions, Fourier coefficients of half-integral weight modular forms, Rubin's congruences involving the special values of L-functions of elliptic curves with complex multiplication, and many other problems require congruence properties of the generalized Bernoulli numbers (see [16]-[18], [12], [29], [3], etc.). The first steps in this direction can be found in the papers of...

In the $p^n$-th cyclotomic field ${\mathbb{Q}}_{p^n},p$ a prime number, $n\in \mathbb{N}$, the prime $p$ is totally ramified and the only ideal above $p$ is generated by ${\omega }_n={\zeta }_{p^n}-1$, with the primitive $p^n$-th root of unity ${\zeta }_{p^n}=e^{\frac{2\pi i}{p^n}}$. Moreover these numbers represent a norm coherent set, i.e. ${\mathbf{\text{N}}}_{{\mathbb{Q}}_{p^{n+1}}}/{\mathbb{Q}}_{p^n}\left({\omega }_{n+1}\right)={\omega }_n$. It is the aim of this article to establish a similar result for the ray class field $K_{{𝔭}^n}$ of conductor ${𝔭}^n$ over an imaginary quadratic number field $K$ where ${𝔭}^n$ is the power of a prime ideal in $K$. Therefore the exponential function has to be replaced by a suitable elliptic function....

It is well known by results of Golod and Shafarevich that the Hilbert $2$-class field tower of any real quadratic number field, in which the discriminant is not a sum of two squares and divisible by eight primes, is infinite. The aim of this article is to extend this result to any real abelian $2$-extension over the field of rational numbers. So using genus theory, units of biquadratic number fields and norm residue symbol, we prove that for every real abelian $2$-extension over $\mathbb{Q}$ in which eight primes...

Let $𝕜=\mathbb{Q}\left(\sqrt{2},\sqrt{d}\right)$ be an imaginary bicyclic biquadratic number field, where $d$ is an odd negative square-free integer and ${𝕜}_2^{\left(2\right)}$ its second Hilbert $2$-class field. Denote by $G=\mathrm{Gal}({𝕜}_2^{\left(2\right)}/𝕜)$ the Galois group of ${𝕜}_2^{\left(2\right)}/𝕜$. The purpose of this note is to investigate the Hilbert $2$-class field tower of $𝕜$ and then deduce the structure of $G$.