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Effective Galois theory

Published online by Cambridge University Press:  12 March 2014

Peter La Roche
Peter La Roche*
Affiliation:
Cornell University, Ithaca, New York 14850
*
Applied Mathematics Division, Department of Scientific and Industrial Research, Wellington, New Zealand
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Extract

Krull [4] extended Galois theory to arbitrary normal extensions, in which the Galois groups are precisely the profinite groups (i.e. totally disconnected, compact, Hausdorff groups). Metakides and Nerode [7] produced two recursively presented algebraic extensions K ⊂ F of the rationals such that F is abelian, F is of infinite degree over K, and the Galois group of F over K, although of cardinality c, has only one recursive element (viz. the identity). This indicated the limits of effectiveness for Krull's theory. (The Galois theory of finite extensions is completely effective.) Nerode suggested developing a natural effective version of Krull's theory (done here in §1).

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 (done here in §2). 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 46 , Issue 2 , June 1981 , pp. 385 - 392
Copyright
Copyright © Association for Symbolic Logic 1981

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References

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