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 $L$.

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 definition 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'= 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:

#If $K$ is finite, then so is $L$, and any primitive root of $L$ will generate the field extension.
#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 $L$.Lemma 9.19.1 (Primitive element)
The Stacks project. Accessed on line July 19, 2023.

History

This theorem was found and proven in 1910 by Ernst Steinitz.{{Cite journal, last=Steinitz, first=Ernst, date=1910, title=Algebraische Theorie der Körper., url=https://gdz.sub.uni-goettingen.de/id/PPN243919689_0137?tify=%7B%22view%22:%22info%22,%22pages%22:%5B171%5D%7D, journal=Journal für die reine und angewandte Mathematik, language=de, volume=1910, issue=137 , pages=167–309, doi=10.1515/crll.1910.137.167, s2cid=120807300 , issn=1435-5345, url-access=subscription

References

Field (mathematics)
Theorems in abstract algebra