Exercise 31, § 2

a) Let K be a field whose algebraic closure E is an extension of K of prime degree q. Show that K is perfect (V, p. 7). b) Show that q is distinct from the characteristic of K (V, p. 162, Ex. 9).

c) Show that K contains the q-th roots of unity, and that E is the splitting field of an irreducible polynomial of the form $X^q - a$ over K ; deduce that $q = 2$ (otherwise, use Ex. 5, V, p. 161 to deduce that $X^{q^2} - a$ would be irreducible). Show also that a is a square in K (V, loc. cit.), that $-1$ is not a square in K, and that $E = K(i)$ ($i^2 = -1$).

d) Now suppose that K is such that its algebraic closure E has arbitrary finite degree $\neq 1$ over K. Show that $i \notin K$ and that $E = K(i)$ (if $E \neq K(i)$, show by Galois theory that there would exist a field F such that $K(i) \subset F \subset E$, and that E has prime degree over F ; then apply c)). Deduce that K is a maximal ordered field (under some suitable ordering) (show by induction that any sum of n squares in K is a square in K ; to prove that $a^2 + b^2$ is a square, consider a square root $x + iy$ of $a + ib$ in $K(i)$).