Cyclic Algebras and Construction of Some Galois Modules

Let p be a prime and suppose that K/F is a cyclic extension of degree p with group G. Let J be the FpG-module K/K of pth-power classes. In our previous paper we established precise conditions for J to contain an indecomposable direct summand of dimension not a power of p. At most one such summand exists, and its dimension must be p +1 for some 0 ≤ i < n. We show that for all primes p and all 0 ≤ i < n, there exists a field extension K/F with a summand of dimension p + 1. Let p be a prime and K/F a cyclic extension of fields of degree p with Galois group G. Let K be the multiplicative group of nonzero elements of K and J = J(K/F ) := K/K be the group of pthpower classes of K. We see that J is naturally an FpG-module. In our previous paper [MSS] we established the decomposition of J into indecomposables, as follows. For i ∈ N let ξpi denote a primitive pth root of unity, and for 0 ≤ i ≤ n let Ki/F be the subextension of degree p, with Gi = Gal(Ki/F ). We adopt the convention that for all i, {0} is a free FpGi-module. Theorem. [MSS, Theorems 1, 2, and 3] Suppose • F does not contain a primitive pth root of unity or • p = 2, n = 1, and −1 / ∈ NK/F (K),