TY - JOUR
T1 - Partially normal radical extensions of the rationals
AU - Gay, David A.
AU - McDaniel, Andrew
AU - Vélez, William Yslas
N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.
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
VL - 72
SP - 403
EP - 417
JO - Pacific Journal of Mathematics
JF - Pacific Journal of Mathematics
SN - 0030-8730
IS - 2
ER -