# Is there an extension field of degree infinite has no intermediate field?

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 Answer

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 $$E= F(\alpha)$$.

• 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 – reuns Feb 20 at 18:58
• Thanks everyone :) All comments helpful for me. – Primavera Feb 22 at 8:07