Exercise 13, § 14

In an algebraic closure $\Omega$ of $\mathbf{Q}(X)$ consider two pure transcendental extensions $E = \mathbf{Q}(X)$ and $F = \mathbf{Q}(X + i)$ (where $i^2 = -1$) of the field Q. Show that $E \cap F = \mathbf{Q}$ but that E and F are not algebraically disjoint over Q.