We show that the S-Euclidean minimum of an ideal class is a rational number, generalizing a result of Cerri. In the proof, we actually obtain a slight refinement of this and give some corollaries which explain the relationship of our results with Lenstra's notion of a norm-Euclidean ideal class and the conjecture of Barnes and Swinnerton-Dyer on quadratic forms. In particular, we resolve a conjecture of Lenstra except when the S-units have rank one. The proof is self-contained but uses ideas from...

We study the capitulation of $2$-ideal classes of an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $𝕜=\mathbb{Q}(\sqrt{2pq},\mathrm{i})$, where $\mathrm{i}=\sqrt{-1}$ and $p\equiv -q\equiv 1(mod4)$ are different primes. For each of the three quadratic extensions $𝕂/𝕜$ inside the absolute genus field ${𝕜}^{(*)}$ of $𝕜$, we determine a fundamental system of units and then compute the capitulation kernel of $𝕂/𝕜$. The generators of the groups ${\mathrm{Am}}_s(𝕜/F)$ and $\mathrm{Am}(𝕜/F)$ are also determined from which we deduce that ${𝕜}^{(*)}$ is smaller than the relative genus field ${(𝕜/\mathbb{Q}\left(\mathrm{i}\right))}^{*}$. Then we prove that each...

Let $K$ be an imaginary cyclic quartic number field whose 2-class group is of type $(2,2,2)$, i.e., isomorphic to $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$. The aim of this paper is to determine the structure of the Iwasawa module of the genus field $K^{(*)}$ of $K$.

From a paper by A. Angelakis and P. Stevenhagen on the determination of a family of imaginary quadratic fields $K$ having isomorphic absolute Abelian Galois groups $A_K$, we study any such issue for arbitrary number fields $K$. We show that this kind of property is probably not easily generalizable, apart from imaginary quadratic fields, because of some $p$-adic obstructions coming from the global units of $K$. By restriction to the $p$-Sylow subgroups of $A_K$ and assuming the Leopoldt conjecture we show that the...

Let $K={𝔽}_q\left(T\right)$ be the rational function field over a finite field of $q$ elements. For any polynomial $f\left(T\right)\in {𝔽}_q\left[T\right]$ with positive degree, denote by ${\Lambda }_f$ the torsion points of the Carlitz module for the polynomial ring ${𝔽}_q\left[T\right]$. In this short paper, we will determine an explicit formula for the analytic class number for the unique subfield $M$ of the cyclotomic function field $K\left({\Lambda }_P\right)$ of degree $k$ over ${𝔽}_q\left(T\right)$, where $P\in {𝔽}_q\left[T\right]$ is an irreducible polynomial of positive degree and $k>1$ is a positive divisor of $q-1$. A formula for the analytic class number for the...