Non-simplicity of \(\mathbb{F}_p(x,y)/\mathbb{F}_p(x^p,y^p)\) (D&F 14.4.6)

Prove that \(\mathbb{F}_p(x,y)/\mathbb{F}_p(x^p,y^p)\) is not a simple extension by explicitly exhibiting an infinite number of infinite subfields.

Let \(F = \mathbb{F}_p(x^p,y^p)\), and consider the extensions \(F(x^{pn+1}+y)\), where \(n \in \mathbb{Z}^+ \cup \{0\}\). They are degree \(p\) over \(F\) (due to the lone \(y\) term) and so are proper subfields of the degree \(p^2\) extension \(\mathbb{F}_p(x,y)\).

Suppose \(F(x^{pi+1}+y) = F(x^{p(i+k)+1}+y)\) where \(i \ge 0, k > 0\) (denote the field \(K\)). Then \(x^{p(i+k)+1}-x^{pi+1} = x^{pi+1}(x^{pk}-1) \in K\), and dividing by \(x^{pi}(x^{pk}-1)\) (an element of \(F\)) gives \(x \in K\). In particular, \(x^{pi+1} \in K\) so \(y \in K\), a clear contradiction. Hence the extensions \(F(x^{pn+1}+y)\) are all distinct, and there are infinitely many of them.