Soient $p_1\equiv p_2\equiv 1(mod8)$ des nombres premiers tels que, $\left(\frac{p_1}{p_2}\right)=-1$ et $\left(\frac{2}{a+b}\right)=-1$, où $p_1{p}_2=a^2+b^2$. Soient $i=\sqrt{-1}$, $d=p_1{p}_2$, $𝕜=\mathbb{Q}(\sqrt{d},i)$, ${𝕜}_2^{\left(1\right)}$ le 2-corps de classes de Hilbert de $𝕜$ et ${𝕜}^{(*)}=\mathbb{Q}(\sqrt{p_1},\sqrt{p_2},i)$ le corps de genres de $𝕜$. La 2-partie $C_{𝕜,2}$ du groupe de classes de $𝕜$ est de type $(2,2,2)$, par suite ${𝕜}_2^{\left(1\right)}$ contient sept extensions quadratiques non ramifiées ${𝕂}_j/𝕜$ et sept extensions biquadratiques non ramifiées ${𝕃}_j/𝕜$. Dans ce papier on s’intéresse à déterminer ces quatorze extensions, le groupe $C_{𝕜,2}$ et à étudier la capitulation des 2-classes d’idéaux de $𝕜$ dans ces extensions.

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree $\mathsf{c}\left(H\right)$ of $H$ is the smallest integer $N$ with the following property: for each $a\in H$ and each two factorizations $z,z^{\prime }$ of $a$, there exist factorizations $z=z_0,...,z_k=z^{\prime }$ of $a$ such that, for each $i\in [1,k]$, $z_i$ arises from $z_{i-1}$ by replacing at most $N$ atoms from $z_{i-1}$ by at most $N$ new atoms. Under a very mild condition...

1. Introduction. Let k be a totally real number field. Let p be a fixed prime number and ℤₚ the ring of all p-adic integers. We denote by λ=λₚ(k), μ=μₚ(k) and ν=νₚ(k) the Iwasawa invariants of the cyclotomic ℤₚ-extension $k_{\infty }$ of k for p (cf. [10]). Then Greenberg’s conjecture states that both λₚ(k) and μₚ(k) always vanish (cf. [8]). In other words, the order of the p-primary part of the ideal class group of kₙ remains bounded as n tends to infinity, where kₙ is the nth layer of $k_{\infty }/k$. We know by the Ferrero-Washington...

Let H be a Krull monoid with class group G. Then every nonunit a ∈ H can be written as a finite product of atoms, say $a=u_1·...·u_k$. The set (a) of all possible factorization lengths k is called the set of lengths of a. If G is finite, then there is a constant M ∈ ℕ such that all sets of lengths are almost arithmetical multiprogressions with bound M and with difference d ∈ Δ*(H), where Δ*(H) denotes the set of minimal distances of H. We show that max Δ*(H) ≤ maxexp(G)-2,(G)-1 and that equality holds if every...

Let $p$ and $q$ be two different prime integers such that $p\equiv q\equiv 3(mod8)$ with $(p/q)=1$, and $l$ a positive odd square-free integer relatively prime to $p$ and $q$. In this paper we investigate the unit groups of number fields $𝕃=\mathbb{Q}(\sqrt{2},\sqrt{p},\sqrt{q},\sqrt{-l})$.