Let $F = F_0(t_1^p, t_2^p)$ and $L = F_0(t_1, t_2)$. Show that (a) if $\theta \in L - F$ then $[F(\theta) : F] = p$, and (b) There exist infinitely many fields $K$ satisfying $F < K < L$.
Here, I believe that $p$ should be the characteristic of $F$ that is greater than $0$, for otherwise the problem would make little sense.
I am not sure how to approach either of the problem here. For the second one, I was thinking of having $K$ something of the form $F_0(t_1^p, t_2^k)$ with $(k,p) = 1$, but am not sure with how to verify that such $K$ satisfies the condition - my intuition is only that I must somehow temper with the powers of $t_1$ and $t_2$ in regards to the prime $p$. For the part $(a)$ one way of showing it would be to show that $\{ 1, \theta, \theta^2,...,\theta^{p-1} \}$ would be a basis, though this of course may not be true.
Any help would be great.
Surely we are in characteristic $p$.
If $\theta\in L$ then $\theta^p\in F$. Let $a=\theta^p$. Then $\theta$ is a zero of $X^p-a=(X-\theta)^p$. If $\theta\notin L$ then $X^p-a$ is irreducible over $F$. Otherwise $X^p-a$ has a factor over $F$ of the form $(X-\theta)^k$ with $1\le k<p$ and then $\theta^k\in F$. As $\theta^p\in F$, it follows that $\theta\in F$.
We conclude $|F(\theta):F|=p$.