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 ${\displaystyle L}$.

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