Steinitz's theorem (field theory)

In field theory, Steinitz's theorem states that a finite extension of fields ${\displaystyle L/K}$ is simple if and only if there are only finitely many intermediate fields between ${\displaystyle K}$ and ${\displaystyle L}$.

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

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

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

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

This theorem was found and proven in 1910 by Ernst Steinitz.^[2]