Steinitz's theorem (field theory)

In field theory, Steinitz's theorem states that a finite extension of fields $L/K$ is simple if and only if there are only finitely many intermediate fields between $K$ and $L$. (If $K$ is infinite, the finiteness of the extension $L/K$ follows in either case, cf. the proof below, and may thus be omitted from the theorem statement.)

Proof

Suppose first that $L/K$ is simple, that is to say $L=K(\alpha )$ for some $\alpha \in L$. Let $M$ be any intermediate field between $L$ and $K$, and let $g$ be the minimal polynomial of $\alpha$ over $M$. Let $M'$ be the field extension of $K$ generated by all the coefficients of $g$. Then $M'\subseteq M$ by def. of the minimal polynomial, but the degree of $L$ over $M'$ is (like that of $L$ over $M$) simply the degree of $g$. Therefore, by multiplicativity of degree, $[M:M']=1$ and hence $M=M'$.

But if $f$ is the minimal polynomial of $\alpha$ over $K$, then $g|f$, and since there are only finitely many divisors of $f$, the first direction follows.

Conversely, if the number of intermediate fields between $L$ and $K$ is finite, we distinguish two cases:

1. If $K$ is finite, then so is $L$, and any primitive root of $L$ will generate the field extension.
2. If $K$ is infinite, then each intermediate field between $K$ and $L$ is a proper $K$-subspace of $L$, and their union can't be all of $L$. Thus any element outside this union will generate $L$.

History

This theorem was found and proven in 1910 by Ernst Steinitz.