Studying a prime field, Prime fields has no proper subfields.
At this point, i have some question.
Suppose that $F$ is a field and $E$ is an extension of $F$. Suppose further that there is no intermediate field between $F$ and $E$.
Then, is it true that
1) Both $F$ and $E$ equals some prime fields?
2) $E$ is a simple extension of $F$?
or is there an extension of $F$ of degree infinite?
Give some advice. Thank you!
1) No: for example you can take any extension of degree $2$ (e.g. $\Bbb C / \Bbb R$). These need not to be prime fields.
2) Yes. Take any element $\alpha \in E \setminus F$. Then $$F \subsetneq F( \alpha) \subseteq E$$ by our hypothesis this means that $F( \alpha) =E$.
3) Such an extension cannot have infinite degree. Indeed, being a simple extension by 2), there are two cases:
3.1) if $\alpha$ is algebraic over $F$, it has finite degree $n$, then necessarily $$[E:F] = [F ( \alpha) : F] = n$$
3.2) if $\alpha$ is transcendental, then $F(\alpha^2)$ is an intermediate field between $F$ and $E= F(\alpha)$.