A formula for the S-class number of an algebraic torus

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

We obtain a formula for the S-class number of an algebraic torus defined over a number field in terms of the étale 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.

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Proof. Using that (Z/p)× is a cyclic group of order p − 1 (i.e. the existence of primitive roots), we see that there is a square root of −1 (that is, a non-trivial fourth root of 1) in (Z/p)× if and only if p ≡ 1 mod 4. Suppose now that p ≡ −1 mod 4, and suppose that α and β are two elements of Z[i] such that p|αβ. Then p = N(p)|N(α)N(β), and so (after relabelling if necessary) we may assume th...

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ژورنال

عنوان ژورنال: Journal of Number Theory

سال: 2017

ISSN: 0022-314X

DOI: 10.1016/j.jnt.2017.06.010