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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!

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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 $E= F(\alpha)$.

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    $\begingroup$ Also for any $n$ let $F(\beta)/F$ Galois such that $Gal(F(\beta)/F) = S_n$ and $H = Stab(\beta) \cong S_{n-1}$ then $ F^H/F$ is of degree $n$ and has no intermediate field $\endgroup$ – reuns Feb 20 at 18:58
  • $\begingroup$ Thanks everyone :) All comments helpful for me. $\endgroup$ – Primavera Feb 22 at 8:07

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