Exercise 15, § 1

Let K be a commutative field of characteristic 0 containing all the roots of unity and such that there exist Galois extensions of K whose Galois group is not solvable (cf. for example Algebra, Chapter V, Appendix I, no. 1, Proposition 2). Let Ω be an algebraic closure of K and let Ω_0 be the set of elements z ∈ Ω such that the Galois extension of K generated by z in Ω admits a solvable Galois group.

(a) Show that Ω_0 is a subgroup of Ω distinct from Ω (cf. Algebra, Chapter V, § 10, no. 4, Theorem 1).

(b) Let A be the subring of Ω[X] consisting of the polynomials P such that P(0) ∈ R,. Show that A is not integrally closed but that every element of its field of fractions Ω(X) a power of which belongs to A is itself in A.