On the $2$-class group of some number fields with large degree

TL;DR

This paper investigates the structure of the 2-class group in certain cyclotomic extensions of number fields, identifying conditions for odd class numbers and computing 2-class group ranks in specific cases.

Contribution

It determines all fields of the form $L_{m, d}$ with odd class number and computes the 2-class group rank using cyclotomic $Z_2$-extensions for particular prime divisors.

Findings

Identifies all $L_{m, d}$ with odd class number.

Calculates 2-class group rank for primes $mod 8$.

Provides explicit conditions for class number properties.

Abstract

Let $d$ be an odd square-free integer, $m\geq 3$ any integer and $L_{m, d}:=\mathbb{Q}(\zeta_{2^m},\sqrt{d})$. In this paper, we shall determine all the fields $L_{m, d}$ having an odd class number. Furthermore, using the cyclotomic $\mathbb{Z}_2$-extensions of some number fields, we compute the rank of the $2$-class group of $L_{m, d}$ whenever the prime divisors of $d$ are congruent to $3$ or $5\pmod 8$.