Partially normal radical extensions of the rationals

David A. Gay, Andrew McDaniel, William Yslas Velez

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If K is a field and char K\n, then any binomial xn - b ϵ K[x] has the property that K(a) is its splitting field for any root a iff a primitive nth root of unity an is an element of K. Thus, if aneK, any irreducible binomial xn - b ϵ K[x] is automatically normal. Similar nice results about binomials xn – b (Kummer theory comes to mind) can be obtained with the assumption l ϵ K. In this paper, without assuming the appropriate roots of unity are in K, one asks: what are the binomials xm - b ϵ K[x] having the property that K(a) is its splitting field for some root al Such binomials are called partially normal. General theorems are obtained in case K is a real field. A complete list of partially normal binomials together with their Galois groups is found in case K–Q, the rational numbers.

Original languageEnglish (US)
Pages (from-to)403-417
Number of pages15
JournalPacific Journal of Mathematics
Issue number2
Publication statusPublished - 1977


ASJC Scopus subject areas

  • Mathematics(all)

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