Exercise 6, § 10

Let N be a Galois extension of a field K, of infinite degree, and let $\rho, a$ be two distinct elements of $\mathrm{Gal}(N/K)$. If J is the set of $x \in N$ such that $\rho(x) \neq \sigma(x)$, show that $K(J)$ is an extension of K of infinite degree. (Observe that a field cannot be the union of two of its subfields unless it is equal to one of them.)