tower of simple extensions of $\mathbb Z$$_p$ for which the full extension will not be simple

Question

Let $s$ and $t$ be independent variables and let $p$ be a prime. Show that in the tower

$\mathbb Z$$_p$$(s^p,t^p)\lt\mathbb Z$$_p$$(s,t^p)\lt\mathbb Z$$_p$$(s,t)$

each step is simple but the full extension is not.

In fact it is easy to see that each step is simple, since we adjoin in each step only one element. In the first step we adjoin $s$ and in the next step we adjoin $t$. So the steps are simple extensions.

Of course the full extension will not be simple but how shall I explain or prove it?

Thanks for any detailed help..

Answer 1

Be careful that when you consider $K\lt F$, what you have in K are polynomials over polynomials like $f(s^p,t^p)/g(s^p,t^p)$.

And, adjoining only $s$, you need to show you can not obtain $t$ (which indeed is in $F$) from such polynomials. And similarly, adjoining only $t$, you need to show you can not obtain $s$ with those polynomials either (which again should be in $F$).

This proves that $F$ over $K$ is not a simple extension.

Answer 2

Let the fields be (from left to right) $K$, $L$ and $F$ respectively.

Hint: Show that for all elements $z\in F$ we have $z^p\in K$. Show that this implies that any simple extension $K(z)$ of $K$ inside $F$ has degree $[K(z):K]=p$ (or $1$). But also show that $[F:K]=p^2$.