An action of a Galois group on a tensor product

Abstract

Let K be a Galois extension of a field F with G=Gal(K/F).Let L be an extension of F such that Kotimes_F L= N_1 oplus N_2 oplus cdots oplus N_k with corresponding primitive idempotents e_1,e_2,cdots,e_k, where N_i's are fields. Then G acts on e_1,e_2,cdots,e_k transitively and Gal(N_1/K)cong sigma in G~|~sigma(e_1)=e_1. And, let R be a commutative F-algebra, and let P be a prime ideal of R. Let T=Kotimes_F R, and suppose there are only finitely many prime ideals Q_1,Q_2,cdots,Q_k of T with Q_i cap R=P. Then G acts transitively on Q_1,Q_2,cdots,Q_k, and Gal(qf(T/Q_1)/qf(R/P)) cong sigma in G~|~sigma(Q_1)=Q_1 where qf(T/Q_1) is the quotient field of T/Q_1.