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Fields countably generated over a proper subfield

Published online by Cambridge University Press:  17 April 2009

James K. Deveney  and
Joe Yanik
James K. Deveney
Affiliation:
Department of Mathematical Sciences, Virginia Commonwealth University, Richmond, Virginia 23284., U.S.A.
Joe Yanik
Affiliation:
Department of Mathematical Sciences, Virginia Commonwealth University, Richmond, Virginia 23284., U.S.A.
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Abstract

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For an arbitrary field K there are two related questions that can be asked:

(1) Is there a proper subfield, L, of K such that K is countably generated over L?

(2) Given a proper subfield M of K is there a proper subfield, L, of K containing M such that K is countably generated over L?

We give an affirmative answer to (1) in characteristic p ≠ 0 and provide counterexamples to (2) for arbitrary characteristic ≠ 2.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 33 , Issue 2 , April 1986 , pp. 219 - 226
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Bialynicki-Birula, A., “On subfields of countable codimension”, Proc. Amer. Math. Soc. 35 (1972), 354–356.CrossRefGoogle Scholar
[2]Jacobson, N., Lectures in abstract algebra, Vol. III. Theory of fields and Galois theory, (Von Nostrand, Princeton, N.J., 1964).Google Scholar