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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

$ p$th powers of distinguished subfields


Author: Nicholas Heerema
Journal: Proc. Amer. Math. Soc. 55 (1976), 287-292
DOI: https://doi.org/10.1090/S0002-9939-1976-0392949-X
MathSciNet review: 0392949
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Abstract: Let $ k \supset s \supset {k_0}$ be fields of characteristic $ p \ne 0,k/{k_0}$ finitely generated and $ s$ a distinguished subfield. The field $ {k_0}({k^{(n)}}) = \{ x \in k\vert{x^{{p^t}}} \in {k_0}({k^{{p^{n + t}}}}){\text{ for some }}t \geqslant 0\} $ has $ {k_0}({s^{{p^n}}})$ as a distinguished subfield and is maximal in $ k$ with respect to this property. Let $ \overline k _0^s$ and $ {\bar k_0}$ be, respectively, the separably algebraic closure and the algebraic closure of $ {k_0}$ in $ k$. Then $ {\overline k _0} = { \cap _k}{k_0}({k^{(n)}})$. Also $ \overline k _0^s = {\bar k_0}$ if and only if $ {k_0}({k^p}) \supset {k_0}({k^{(n)}})$ for some $ n$. For $ n$ large $ {k_0}({k^{(n)}}) = {\overline k _0}({k^{{p^n}}})$. The sequence $ {\{ {[{k_0}({k^{(n)}}):{k_0}]_i}\} _n}$ is decreasing, descending from $ {[k:{k_0}]_i}\;{\text{to }}{[\overline {k:} {k_0}]_i}$ in a finite number of steps. Examples are given which show: (1) that $ {k_0}({k^{(n)}})$ may have distinguished subfields not of the form $ {k_0}({s^{{p^n}}})$; and, (2) how to construct $ k/{k_0}$ so that the sequence $ \{ [{k_0}{({k^{(n)}}:{k_0}]_i}\} $ has preassigned values.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0392949-X
Article copyright: © Copyright 1976 American Mathematical Society