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The main purpose of this article in some sense is to illustrate the manner in which the classical methods of Galois descent work to impose constraints on the algebraic behavior of local rings (for contrast, see A. Grothendieck, “Seminaire de Geometrie Algebrique” 1962 I.H.E.S., fasc. I and II, 1963) . By way of example, we show in Section 2 that a factorial local domain which arises as a normal extension (see definition) of a regular local ring is necessarily a complete intersection. This is in sharp contrast to the situation for general normal domains (e.g., see M.-J, Bertin, C.R. Acad. Sci. Paris 264 (1967) , 653–656; P. Griffith, J. Pure Appl. Algebra 7 (1976) , 303–315; and R. Fossum and P. Griffith, Ann. Sci. Ecole Norm. Sup. (4) 8 (1974) , 189–200).

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