This is a question which came to my mind, when seeing A quick question on transcendence
Suppose $F$ is a field of characteristic $p$. Then the set of $p^n$-powers of the elements of $F$ is again a field which I denote by $F^n$ and is a subfield of $F$. Now let $d_n=[F:F^n]$.
My question is, what can be said about $F$ if we know the sequence $(d_n)_n$. Of course for finite fields $d_n$ is monotonically increasing and eventually constant. The constant will be the index degree of $F$ over the prime field $F_p$ and thus $F$ is determined up to isomorphism.
So what happens in general. Of course $d_n$ is monotonically increasing. In general $d_n$ will not be finite. This can be seen by taking $F=F_p(t_1,t_2,\ldots)$ where $t_i$ is transcendental over $F_p(t_1,t_2,\ldots,t_{i-1})$.
So now my questions:
Can it happen that $d_1$ is finite but $d_n$ is infinite for some $n$?
Under the assumption that $(d_n)_n$ is finite and the same for a field $F_1$ and $F_2$. Are these fields necessarily isomorphic?