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..