Sur la capitulation des 2-classes d'id\'eaux du corps Q(\sqrt{2p_1p_2}, i)

TL;DR

This paper investigates the capitulation of 2-classes in certain number fields formed by adjoining square roots of primes and imaginary units, focusing on the behavior of class groups in unramified extensions.

Contribution

It provides a detailed analysis of the capitulation of 2-classes in specific bi-quadratic and cyclotomic extensions of quadratic fields with primes satisfying particular congruences.

Findings

Capitulation of 2-classes in the extensions $K_1$, $K_2$, $K_3$, and $k^{(*)}$.

Conditions under which 2-classes capitulate in these unramified extensions.

Structural insights into the 2-part of the class group of the field $k$.

Abstract

Let $p_1$ and $p_2$ be two primes such that $p_1\equiv p_2\equiv1 \pmod4$ and at least two of the three elements $\{(\frac{2}{p_1}), (\frac{2}{p_2}), (\frac{p_1}{p_2})\}$ are equal to -1. Put $i=\sqrt{-1}$, $d=2p_1p_2$ and $k =Q(\sqrt{d}, i)$. Let $k_2^{(1)}$ be the Hilbert 2-class field of $k$ and $k^{(*)}=Q(\sqrt{p_1},\sqrt{p_2},\sqrt 2, i)$ be its genus field. Let $C_{k,2}$ denote the 2-part of the class group of $k$. The unramified abelian extensions of $k$ are $K_1=k(\sqrt{p_1})$, $K_2=k(\sqrt{p_2})$, $K_3=k(\sqrt{2})$ and $k^{(*)}$. Our goal is to study the capitulation problem of the 2-classes of $k$ in these four extensions.