Chapter 13

Let \(p\) be a prime, let \(J=\mathbb{Z}_{p}(\alpha)\), where \(\alpha\) is transcendental over \(\mathbb{Z}_{p}\), and let \(K=J(\beta)\), where \(\beta\) is transcendental over \(J\). Let \(L: K\) be a splitting field extension for \(\left(x^{p}-\alpha\right)\left(x^{p}-\beta\right)\). (i) Show that \([L: K]=p^{2}\). (ii) Show that if \(\gamma \in L\) then \(\gamma^{p} \in K\). (iii) Show that \(L: K\) is not simple. (iv) In the case where \(p=2\), find all the intermediate fields \(L: M: K\).

Suppose that \(L: K\) is a Galois extension with Galois group \(\left\\{\sigma_{1}, \ldots, \sigma_{n}\right\\}\) and that \(\alpha \in L\). Show that \(L=K(\alpha)\) if and only if \(\left(\sigma_{1}(\alpha), \ldots, \sigma_{n}(\alpha)\right)\) is a basis for \(L\) over \(K\).

Suppose that \(K(t): K\) is a simple transcendental extension. Show that there are infinitely many intermediate fields.

Suppose that \(L: K\) is a finite separable extension and that \(M: L\) is a finite simple extension. Show that \(M: K\) is a simple extension.