Let 77 be a finite group and denote by M π the class of all (finitely generated Z-free) 77-modules. In the previous paper [3] we defined an equivalence relation in M π and constructed the abelian semigroup T(77) by giving an addition to the set of all equivalence classes in M π . The investigation of the semigroup T(Π) seems interesting and important, because this gives a classification of the function fields of algebraic tori defined over a field k which split over a Galois extension of k with group 77.The purpose of this paper is to obtain information on the structure of the semigroup Γ(77).We will recall the definitions given in Let k be a field. Let K be a Galois extension of k with group s 77 and let M be a 77-module with a Z-free basis {u u u 2 , ,w n }. Define the action on the rational function field K(X 19 X 2 , , X n ) with n variables X lf X 2 > --,X n over K by putting, for each σe77 and 1 <^ i<>n, σ{Xd = \\ n j=1 X^ when σ Ut = Σ?=i m *Λ > mtjeZ, and denote by K(M) K(X l9 X 2 , • , X Λ ) with this action of 77. It is well known ([7]) that there is a duality between the category of all algebraic tori defined over k which split over K and the category of all 77-modules. In fact, if T is an algebraic torus defined over k which splits over K, then the character
There are some errors in Theorems 3.3 and 4.2 in [2]. In this note we would like to correct them.1) In Theorem 3.3 (and [IV]), the condition (1) must be replaced by the following one;(1) П is (i) a cyclic group, (ii) a dihedral group of order 2m, m odd, (iii) a direct product of a cyclic group of order qf, q an odd prime, f ≧ 1, and a dihedral group of order 2m, m odd, where each prime divisor of m is a primitive qf-1(q — 1)-th root of unity modulo qf, or (iv) a generalized quaternion group of order 4m, m odd, where each prime divisor of m is congruent to 3 modulo 4.
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