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 $E= F(\alpha)$.

  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.