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 ${\displaystyle L}$.

Proof

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]

History

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

References

1. Lemma 9.19.1 (Primitive element), The Stacks project. Accessed on line July 19, 2023.
2. Steinitz, Ernst (1910). "Algebraische Theorie der Körper". Journal für die reine und angewandte Mathematik (in German). 1910 (137): 167–309. doi:10.1515/crll.1910.137.167. ISSN   1435-5345. S2CID   120807300.