Roots appear in quanta: exercise solutions

Herein are solutions to the exercises given in [1]. Let K be a field, and f(X) ∈ K[X] an irreducible polynomial. The root quantum number rK(f) is the number of roots of f in the stem field K(α), where α is an arbitrary choice of root of f . It was shown that rK(f) is well-defined and divides the degree of f . We start by establishing the connection with Galois theory. Let f be irreducible and separable over K. Let L/K be the splitting field and G the Galois group. Fix a root α, and let H ⊂ G be the subgroup fixing α. A root of f lies in K(α) if and only if it is fixed by H, so rK(f) equals the number of roots fixed by H. Any automorphism of K(α)/K is determined by the image of α, which must be another root of f lying in K(α); conversely, the map sending α to any root of f in K(α) gives rise to an automorphism. Thus rK(f) equals the cardinality of Aut ( K(α)/K ) . Finally, rK(f) equals the index [NG(H) : H] of H in its normalizer, since Galois theory (see below) tells us Aut ( K(α)/K ) ∼= NG(H)/H.