A Formula For the S-Class Number of an Algebraic Torus

TL;DR

This paper derives a new formula for the S-class number of algebraic tori over number fields using cohomology groups, providing alternative proofs for classical class number formulas by notable mathematicians.

Contribution

It introduces a cohomological formula for the S-class number of algebraic tori, connecting it with etale and Galois cohomology, and offers new proofs of existing class number formulas.

Findings

Derived a formula for the S-class number in terms of cohomology groups.

Provided alternative proofs for classical class number formulas.

Connected algebraic tori properties with cohomological invariants.

Abstract

We obtain a formula for the S-class number of an algebraic torus defined over a number field in terms of the etale and Galois cohomology groups of its character module. As applications, we give different proofs of some classical class number formulas of Shyr, Ono, Katayama and Morishita.