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 $K$.

Let K, L be algebraic number fields with K ⊆ L, and $O_K$, $O_L$ their respective rings of integers. We consider the trace map $T=T_{L/K}:L\to K$ and the $O_K$-ideal $T\left(O_L\right)\subseteq O_K$. By I(L/K) we denote the group indexof $T\left(O_L\right)$ in $O_K$ (i.e., the norm of $T\left(O_L\right)$ over ℚ). It seems to be difficult to determine I(L/K) in the general case. If K and L are absolutely abelian number fields, however, we obtain a fairly explicit description of the number I(L/K). This is a consequence of our description of the Galois module structure of $T\left(O_L\right)$ (Theorem 1). The case...

A $p$-adic version of Stark’s Conjecture at $s=1$ is attributed to J.-P. Serre and stated (faultily) in Tate’s book on the Conjecture. Building instead on our previous paper (and work of Rubin) on the complex abelian case, we give a new approach to such a conjecture for real ray-class extensions of totally real number fields. We study the coherence of our $p$-adic conjecture and then formulate some integral refinements, both alone and in combination with its complex analogue. A ‘Weak Combined Refined’ version...

For an algebraic number field $K$ with $3$-class group ${\mathrm{Cl}}_3\left(K\right)$ of type $(3,3)$, the structure of the $3$-class groups ${\mathrm{Cl}}_3\left(N_i\right)$ of the four unramified cyclic cubic extension fields $N_i$, $1\le i\le 4$, of $K$ is calculated with the aid of presentations for the metabelian Galois group ${\mathrm{G}}_3^2\left(K\right)=\mathrm{Gal}\left({\mathrm{F}}_3^2\left(K\right)\right|K)$ of the second Hilbert $3$-class field ${\mathrm{F}}_3^2\left(K\right)$ of $K$. In the case of a quadratic base field $K=\mathbb{Q}\left(\sqrt{D}\right)$ it is shown that the structure of the $3$-class groups of the four $S_3$-fields $N_1,...,N_4$ frequently determines the type of principalization of the $3$-class group of $K$ in $N_1,...,N_4$. This provides...

Let $L/K$ be an extension of algebraic number fields, where $L$ is abelian over $\mathbb{Q}$. In this paper we give an explicit description of the associated order ${𝒜}_{L/K}$ of this extension when $K$ is a cyclotomic field, and prove that $o_L$, the ring of integers of $L$, is then isomorphic to ${𝒜}_{L/K}$. This generalizes previous results of Leopoldt, Chan Lim and Bley. Furthermore we show that ${𝒜}_{L/K}$ is the maximal order if $L/K$ is a cyclic and totally wildly ramified extension which is linearly disjoint to ${\mathbb{Q}}^{\left(m^{\text{'}}\right)}/K$, where $m^{\text{'}}$ is the conductor of $K$.