If K is a field and
charK ∤ 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.
