Vol. 72, No. 2, 1977

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Partially normal radical extensions of the rationals

David Andrew Gay, Andrew McDaniel and William Yslas Vélez

Vol. 72 (1977), No. 2, 403–417

If K is a field and char K n, then any binomial xn b K[x] has the property that K(α) is its splitting field for any root α iff a primitive n-th root of unity ζn is an element of K. Thus, if ζn K, any irreducible binomial xn b K[x] is automatically normal. Similar nice results about binomials xn b (Kummer theory comes to mind) can be obtained with the assumption ζn K.

In this paper, without assuming the appropriate roots of unity are in K, one asks: what are the binomials xm a K[x] having the property that K(α) is its splitting field for some root α? 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.

Mathematical Subject Classification 2000
Primary: 12E10
Secondary: 12A35
Received: 9 December 1976
Published: 1 October 1977
David Andrew Gay
Andrew McDaniel
William Yslas Vélez