The lattice of subfields of a radical extension

MariáAcosta de Orozco, William Yslas Vélez

Research output: Contribution to journalArticle

11 Scopus citations

Abstract

Let xm - a be irreducible over F with char F{does not divide}m and let α be a root of xm - a. The purpose of this paper is to study the lattice of subfields of F(α) F and to this end C( F(α) F, k) is defined to be the number of subfields of F(α) of degree k over F. C( F(α) F, pn) is explicitly determined for p a prime and the following structure theorem for the lattice of subfields is proved. Let N be the maximal normal subfield of F(α) over F and set n = |N : F|, then C( F(α) F, k) = C( F(α) F, (k, n)) = C( N F, (k, n)). The irreducible binomials xs - b, xs - c are said to be equivalent if there exist roots βs = b, γs = a such that F(β) = F(γ). All the mutually inequivalent binomials which have roots in F(α) are determined. Finally these results are applied to the study of normal binomials and those irreducible binomials x2e - a which are normal over F (char F ≠ 2) together with their Galois groups are characterized.

Original languageEnglish (US)
Pages (from-to)388-405
Number of pages18
JournalJournal of Number Theory
Volume15
Issue number3
DOIs
StatePublished - Dec 1982

ASJC Scopus subject areas

  • Algebra and Number Theory

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