Partially normal radical extensions of the rationals

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 x^m - 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.