Exercise 10, § 2

a) Let K be an orderable field and E a Galois extension of K. Show that either E is orderable or there exists an orderable algebraic extension F of K, contained in E, such that E is a quadratic extension of F (use Th. 6 of V, p. 70).

b) Show that the polynomial $X^4 + 2$ is irreducible over $\mathbf{Q}$, and that the extension $K$ of $\mathbf{Q}$ of degree 4, obtained by adjoining a root of this polynomial to $\mathbf{Q}$, contains no orderable subfield other than $\mathbf{Q}$ (find all the subfields of $K$ by Galois theory).