Algebraic Field Extensions This is exercise $6.34$ from Lang's book:

Give an example of a field $K$ which is of degree $2$ over two distinct subfields $E$ and $F$, respectively, but such that $K$ is not algebraic over $E\cap F$.

So, I've nearly proven the existence of such a field, but there is one part I am struggling with, and if anyone could provide an actual example, I'd greatly appreciate it! My proof goes as follows:
Consider the diagram consisting of $k(s,t)$ at the top, $k(t+t^{-1},s)$ on the lower middle left, and $F$ on the lower middle right, where $F$ is the fixed field of the map $\sigma$ with $\sigma(s)=st$ and $\sigma(t)=t^{-1}$. It's easy to see that $x^2-(t+t^{-1})x+1$ is an irreducible polynomial over the left field, hence it is of degree $2$. Now, I've made the claim that $s\notin F$ since $\sigma^n(s)=st^n$ is an infinite chain, but I'm not sure why this works, nor am I sure why $F$ is of degree $2$. After these two steps, then it's clear that the intersection can't contain $s$, hence $k(s,t)$ can't be algebraic over $k(t+t^{-1},s)\cap F$. Can anyone help me fill in these two gaps?

A simpler example: $K=\mathbf{Q}(x)$ has order-$2$ automorphisms $\sigma:x\mapsto -x$ and $\tau:x\mapsto 1-x$ for which $\sigma\tau$ has infinite order (since it maps $x\mapsto x+1$).  I leave it to you to determine $E$ and $F$.
