Let $p$ be an odd prime, $r$ be a primitive root modulo $p$ and $r_i\equiv r^i(modp)$ with $1\le r_i\le p-1$. In 2007, R. Queme raised the question whether the $\ell$-rank ($\ell$ an odd prime $\ne p$) of the ideal class group of the $p$-th cyclotomic field is equal to the degree of the greatest common divisor over the finite field ${𝔽}_{\ell }$ of $x^{(p-1)/2}+1$ and Kummer’s polynomial $f\left(x\right)={\sum }_{i=0}^{p-2}{r}_{-i}{x}^i$. In this paper, we shall give the complete answer for this question enumerating a counter-example.

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.

Given a monic degree $N$ polynomial $f\left(x\right)\in \mathbb{Z}\left[x\right]$ and a non-negative integer $\ell$, we may form a new monic degree $N$ polynomial $f_{\ell }\left(x\right)\in \mathbb{Z}\left[x\right]$ by raising each root of $f$ to the $\ell$th power. We generalize a lemma of Dobrowolski to show that if $m\<n$ and $p$ is prime then $p^{N(m+1)}$ divides the resultant of $f_{p^m}$ and $f_{p^n}$. We then consider the function $(j,k)\mapsto Res(f_j,f_k)mod{p}^m$. We show that for fixed $p$ and $m$ that this function is periodic in both $j$ and $k$, and exhibits high levels of symmetry. Some discussion of its structure as a union of lattices is also given. ...

In this report we study the arithmetic of Rikuna’s generic polynomial for the cyclic group of order $n$ and obtain a generalized Kummer theory. It is useful under the condition that $\zeta \notin k$ and $\omega \in k$ where $\zeta$ is a primitive $n$-th root of unity and $\omega =\zeta +{\zeta }^{-1}$. In particular, this result with $\zeta \in k$ implies the classical Kummer theory. We also present a method for calculating not only the conductor but also the Artin symbols of the cyclic extension which is defined by the Rikuna polynomial.

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...