Prove that $F(x, y)$ is a simple extension over $F(x^2 , y^2)$

Question

Let $ F $ be a field with char$ F \neq 2 $ and let $x$ and $ y $ be two indeterminates. Prove that $ F(x, y) $ is a simple extension over $ F(x^2, y^2) $ generated by $ x + y $.

Answer 1

It's easier to think that $F(\sqrt{u}, \sqrt{v})$ is generated over $F(u,v)$ by $\sqrt{u}+\sqrt{v}$. Write $a = \sqrt{u}+\sqrt{v}$. We want to express $\sqrt{u}$ in terms of $a$ rationally. For that, write $a-\sqrt{u} = \sqrt{v}$, square to get $a^2 - 2\sqrt{u} a + u = v$ and so $$\sqrt{u} = \frac{a^2+u-v}{2 a}$$ that is $$x = \frac{(x+y)^2+ x^2 - y^2}{2(x+y)}$$ as one can check easily