Ideal Class Groups and Galois Modules

Abstract

Given a prime ℓ, to find the group structure of the ℓ-Sylow subgroup of an ideal class group is in general a hard problem. It is easier if the underlying number field is a degree ℓ cyclic extension of another number field. This is the situation we shall assume. Meanwhile, we shall consider a more general setting, namely, ideal class groups with moduli. More precisely, let K be a cyclic extension of a number field k of prime degree ℓ, let f be a nonzero integral ideal of k and let ∞ be a product of certain (possibly no) real places of k. Denote by I(K/k, f) (or I(K/k, f∞)) the group of fractional ideals of K prime to f,and by K _{f ∞} the group of elements in K ^x prime to f, positive at the real places of K dividing ∞, and congruent to elements in k modulo f. Let B(K/k, f ∞) be the subgroup of principal ideals in I(K/k,f) generated by elements in K _{f ∞}. The quotient group T(K/k, f ∞) = I(K/k, f)/B(K/k, f∞) is called the ideal class group of K/k modulo f ∞. Its ℓ-Sylow subgroup L(K/k, f∞) is our main concern. When the modulus foo is trivial, the group is independent of k and will be denoted by L(K).