The rank of the 2-class group of some fields with large degree

TL;DR

This paper investigates the structure of the 2-class group in certain number fields formed by adjoining roots of unity and square roots of specific integers, providing explicit rank computations under particular prime conditions.

Contribution

It offers new explicit formulas for the 2-class group rank of fields generated by roots of unity and square roots, under specific prime congruence conditions.

Findings

Computed the 2-class group rank for fields with specified prime divisors.

Extended previous results to fields with large degree and particular prime congruences.

Provided explicit criteria for the rank based on prime divisibility conditions.

Abstract

Let $n\geq 3$ be an integer and $d$ an odd square-free integer. We shall compute the rank of the $2$-class group of $L_{n,d}:=\mathbb{Q}(\zeta_{2^n},\sqrt{d})$, when all the prime divisors of $d$ are congruent to $\pm 3\pmod 8$ or $9\pmod{16}$.