Galois extensions of prime degree

The analysis of T,(R) is by a “descent” from the ring S, and so we are led to considering T,(S). Since S contains a pth root of unity, namely E, the computation of T,(S) follows easily from standard arguments of Kummer theory. In this section we recall the part of the theory which we need to compute T,(R) [S]. The map Sx + T,(S) sending s to [S[y]/(yp - s), a], where o(y) = EJJ, is a group homomorphism; (S x )” is the kernel, and we obtain an injection j: sx/(s”)p’ T,(S). Given [A, a] E Z”,,(S), consider