Exercise 20, § 14

Let n be an integer, K a field of characteristic p prime to n, possessing primitive n-th roots of unity, $\mu_n \subset K^*$ the subgroup of n-th roots of unity, G a finite commutative group annihilated by n and $A = K^{(G)}$ the group algebra of G (III, p. 446). a) Show that A is a diagonalizable algebra (V, p. 28). (Decompose G into a direct sum of cyclic groups (VII, p. 22).) Deduce that every A-module is a direct sum of modules which are vector K-spaces of dimension 1. b) Let V be an A-module of finite dimension over K, S the symmetric algebra of the vector K-space V and F the field of fractions of S. The group G operates on V, hence on S and F. Assume that G acts faithfully on V. Show that G operates faithfully on F. Calculate [F : F^G]. c) Let $(x_1, ..., x_r)$ be a basis of the vector K-space V consisting of eigenvectors for the action of G. Let $\Gamma$ be the subgroup of F* generated by $x_1, ..., x_n$. Show that $\Gamma$ is a free commutative group and that the sub-K-algebra B of F generated by $\Gamma$ may be identified with $K^{(\Gamma)}$ and is stable under the operation of G. d) For all $x \in \Gamma$, the mapping $\gamma \mapsto \frac{\gamma x}{x}$ is a homomorphism of G into $\mu_n$, giving rise to a homomorphism $0 : \Gamma \to \mathrm{Hom}(G, p.)$. Show that 0 is surjective. Put $\Gamma_1 = \mathrm{Ker}(0)$. Using the elementary divisor theorem (VII, p. 24), show that B is a free $K^{(\Gamma_1)}$-module of rank Card(G). e) By passing to the field of fractions deduce from d) and a) that $F^G$ is the field of fractions of $K^{(\Gamma_1)}$. Deduce that $F^G$ is a pure transcendental extension of K. \*