ON THE LENGTHS OF MAXIMAL CHAINS OF INTERMEDIATE FIELDS IN A FIELD EXTENSION

Abstract

We study two invariants, ν(L/K) and λ(L/K), arising from a field extension L/K. By definition, ν(L/K) is the cardinality of the set of fields F such that K ⊆ F ⊆ L; and λ(L/K) is the supremum of the set of cardinal numbers arising as lengths of chains {Fi } of fields such that K ⊆ Fi ⊆ L. If L can be generated by one element over an infinite field K and 2 ≤ [L : K] = n < ∞, then ν(L/K) ≤ 2 n−2 + 1, with equality if L/K is Galois with the Klein four-group as Galois group. Despite positive results in the case of Abelian Galois groups, λ is not generally additive for finite-dimensional towers; moreover, maximal chains of intermediate fields can exhibit noncatenarian behavior if [L : K] = 12. The theory for infinite-dimensional algebraic extensions is strikingly different. For instance, for each infinite cardinal number N, there exists a field K of cardinality N and a chain of cardinality 2 N consisting of algebraic field extensions of K; in other words, if denotes an algebraic closure of K, then and, a fortiori, . The tendency for λ(L/K) to realize the cardinality of the power set of L persists in other infinitistic contexts, for instance, if |K| and td(L/K) are prescribed infinite cardinal numbers. However, if L/K is any nonalgebraic finitely.generated field extension, then λ(L/K) = N 0.