Abstract

Krull [4] extended Galois theory to arbitrary normal extensions, in which the Galois groups are precisely the profinite groups. Metakides and Nerode [7] produced two recursively presented algebraic extensionsK⊂Fof the rationals such thatFis abelian,Fis of infinite degree overK, and the Galois group ofFoverK, although of cardinalityc, has only one recursive element. This indicated the limits of effectiveness for Krull's theory. Nerode suggested developing a natural effective version of Krull's theory.It is evident from the classical literature that the free profinite group on denumerably many generators can be obtained effectively as the Galois group of a recursive extension of the rationals over a subfield. Nerode conjectured that it could be obtained effectively as the Galois group of the algebraic numbers over a suitable subfield. The case of finitely many generators was done non-effectively by Jarden [3]. The author believes that the denumerable case, as presented in §2, is also new classically. Using this result and the effective Krull theory, every “co-recursively enumerable” profinite group is effectively the Galois group of a recursively enumerable field of algebraic numbers over a recursive subfield.