View source for Steinitz's theorem (field theory)

In [[Field (mathematics)|field theory]], '''Steinitz's theorem''' states that a finite extension of fields <math>L/K</math> is [[simple extension|simple]] if and only if there are only finitely many intermediate fields between <math>K</math> and <math>L</math>.

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

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

Conversely, if the number of intermediate fields between <math>L</math> and <math>K</math> is finite, we distinguish two cases:

#If <math>K</math> is finite, then so is <math>L</math>, and any primitive root of <math>L</math> will generate the field extension.
#If <math>K</math> is infinite, then each intermediate field between <math>K</math> and <math>L</math> is a proper <math>K</math>-subspace of <math>L</math>, and their union can't be all of <math>L</math>. Thus any element outside this union will generate <math>L</math>.<ref>[https://stacks.math.columbia.edu/tag/030N Lemma 9.19.1 (Primitive element)], [[Stacks Project|The Stacks project]].  Accessed on line July 19, 2023.</ref>

== History ==
This theorem was found and proven in 1910 by [[Ernst Steinitz]].<ref name=":0">{{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}}</ref>

== References ==
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[[Category:Field (mathematics)]]
[[Category:Theorems in abstract algebra]]