An isomorphism between the narrow ideal class group of squared ideals of a quadratic number field and the kernel of a homomorphism between cohomology groups for Pell conics

TL;DR

This paper establishes an isomorphism between the narrow ideal class group of squared ideals in quadratic fields and a kernel in cohomology groups related to Pell conics, providing two proofs using algebraic number theory and Pell conic subgroups.

Contribution

It introduces two novel proofs of the isomorphism connecting class groups and cohomology kernels for Pell conics, enhancing understanding of their algebraic structure.

Findings

Proves the isomorphism using algebraic number theory techniques.

Provides a new perspective on Pell conics and class groups.

Clarifies the role of specific subgroups in Pell conics.

Abstract

Two proofs are provided that the narrow ideal class group of squared ideals of a quadratic number field is isomorphic to the kernel of a homomorphism between cohomology groups for Pell conics, Lemmermeyer's obstruction to descent for Pell conics. These proofs make use of a particular subgroup of a Pell conic over algebraic numbers.