TY - JOUR

T1 - Partially normal radical extensions of the rationals

AU - Gay, David A.

AU - McDaniel, Andrew

AU - Vélez, William Yslas

PY - 1977/10

Y1 - 1977/10

N2 - 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.

AB - 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.

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U2 - 10.2140/pjm.1977.72.403

DO - 10.2140/pjm.1977.72.403

M3 - Article

AN - SCOPUS:84972514610

SN - 0030-8730

VL - 72

SP - 403

EP - 417

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

IS - 2

ER -