Galois extensions of prime degree
Over a commutative ring R, the isomorphism classes of \({\mathbb{Z}}/p{\mathbb{Z}}\)-Galois extensions form an abelian group, \(T_ p(R)\), the Harrison group. In this paper, the authors calculate \(T_ p(R)\) for all R in which p is invertible, but which need not contain a primitive \(p-th\) root of 1. The method is to consider \(S=R[X]/(X^{p-1}+\cdots+X+1)\), apply Kummer theory to S to obtain a short exact sequence \(1\to S^{\times}/(S^{\times})^ p\to T_ p(S)\to Pic(S)[p]\to 1,\) apply the functor \((\;)^{\Gamma}\) to this sequence, where \(\Gamma \cong ({\mathbb{Z}}/p{\mathbb{Z}})^*\), and then show that \(T_ p(R)\cong T_ p(S)^{\Gamma}\).
scientific article; zbMATH DE number 4158782
scientific article