Exercise 10, § 11

a) Let K be a field whose algebraic closure is an extension of prime degree q of K. Such that K is perfect and that q must be distinct from the characteristic of K (use Ex. 9).

b) Show that K contains the q-th roots of unity and that $\Omega$ is the splitting field of an irreducible polynomial $X^q - a$ of $K[X]$; deduce that $q = 2$ (in the contrary case deduce from Ex. 5 that the polynomial $X^{q^2} - a$ would be irreducible). Show further that $-a$ is a square in $K$ (loc. cit.), that $-1$ is not a square in $K$, and deduce that $\Omega = K(i)$, where $i^2 = -1$.