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Distinguished Subfields of Intermediate Fields

Published online by Cambridge University Press:  20 November 2018

James K. Deveney  and
John N. Mordeson
James K. Deveney
Affiliation:
Virginia Commonwealth University, Richmond, Virginia
John N. Mordeson
Affiliation:
Virginia Commonwealth University, Richmond, Virginia
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Let L be a finitely generated extension of a field K of characteristic p ≠ 0. If L/K is algebraic, then there is a unique intermediate field S such that

S is just the maximal separable extension of K in L. If L/K is not algebraic, then Dieudonne [4] showed there exist maximal separable extensions D of K in L such that L ⊆ Kp–∞⊗KD. In general, not every maximal separable extension of K in L has the property. Those which do have the property are called distinguished. Kraft [7] established that a maximal separable extension D of K in L is distinguished if and only if [L:D] is as small as possible. If the minimum of the [L:D] is pr, r is called the order of inseparability of L/K, denoted inor (L/K).

Let L1 be an intermediate field of L/K. If L/K is algebraic, then the maximal separable extension S1 of K in L1 is contained in the maximal separable extension S of K in L, and moreover S is separable over S1.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 5 , 01 October 1981 , pp. 1085 - 1096
Copyright
Copyright © Canadian Mathematical Society 1981

References

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