Class number formula for dihedral extensions

TL;DR

This paper provides an algebraic proof of a class number formula for dihedral extensions of number fields, expressing class number ratios via cohomology groups, and derives explicit bounds on these ratios.

Contribution

It introduces a new algebraic proof for the class number formula in dihedral extensions and establishes explicit bounds on class number ratios without deep representation theory.

Findings

Derived explicit bounds on class number ratios in dihedral extensions.

Revealed the ratio of class numbers as a ratio of cohomology group orders.

Unified previous special case formulas within a broader algebraic framework.

Abstract

We give an algebraic proof of a class number formula for dihedral extensions of number fields of degree $2q$, where $q$ is any odd integer. Our formula expresses the ratio of class numbers as a ratio of orders of cohomology groups of units and recovers similar formulas which have appeared in the literature as special cases. As a corollary of our main result we obtain explicit bounds on the (finitely many) possible values which can occur as ratio of class numbers in dihedral extensions. Such bounds are obtained by arithmetic means, without resorting to deep integral representation theory.