Cyclic quartic fields and genus theory of their subfields

Abstract

Let k = Q (√ u) ( u ≠ 1 squarefree), K any possible cyclic quartic field containing k. A close relation is established between K and the genus group of k. In particular: (1) Each K can be written uniquely as K = Q (√ vwη), where η is fixed in k and satisfies η ⪢ 1, (η) = U 2√ u, | U 2| = |(√ u)|, ( v, u) = 1, v ∈ Z is squarefree, w| u, 0 < w < √ u. Thus if u ≠ a 2 + b 2, there is no K ⊃ k. If u = a 2 + b 2 then for each fixed v there are 2 g − 1 K ⊃ k, where g is the number of prime divisors of u. (2) K k has a relative integral basis (RIB) (i.e., O K is free over O k ) iff N( ε 0) = −1 and w = 1, where ε 0 is the fundamental unit of k, (or, equivalently, iff K = Q (√ vε 0√ u), ( v, u) = 1). (3) A RIB is constructed explicitly whenever it exists. (4) disc( K) is given. In particular, the following results are special cases of (2): (i) Narkiewicz showed in 1974 that K k has a RIB if u is a prime; (ii) Edgar and Peterson ( J. Number Theory 12 (1980), 77–83) showed that for u composite there is at least one K ⊃ k having no RIB. Besides, it follows from (4) that the classification and integral basis of K given by Albert ( Ann. of Math. 31 (1930), 381–418) are wrong.