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== History ==
In his First Memoir of 1831, published in 1846,<ref>{{Cite book|last=Neumann|first=Peter M.|url=https://www.worldcat.org/oclc/757486602|title=The mathematical writings of Évariste Galois|date=2011|publisher=European Mathematical Society|isbn=978-3-03719-104-0|___location=Zürich|oclc=757486602}}</ref> [[Évariste Galois]] sketched a proof of the classical primitive element theorem in the case of a [[splitting field]] of a polynomial over the rational numbers. The gaps in his sketch could easily be filled<ref>{{Cite book|last=Tignol|first=Jean-Pierre | author-link=Jean-Pierre Tignol |url=https://www.worldscientific.com/worldscibooks/10.1142/9719|title=Galois' Theory of Algebraic Equations|date=February 2016|publisher=WORLD SCIENTIFIC|isbn=978-981-4704-69-4|edition=2|___location=|pages=231|language=en|doi=10.1142/9719|oclc=1020698655}}</ref> (as remarked by the referee [[Siméon Denis Poisson|Poisson]]) by exploiting a theorem<ref>{{Cite book|last=Tignol|first=Jean-Pierre|url=https://www.worldscientific.com/worldscibooks/10.1142/9719|title=Galois' Theory of Algebraic Equations|date=February 2016|publisher=WORLD SCIENTIFIC|isbn=978-981-4704-69-4|edition=2|pages=135|language=en|doi=10.1142/9719|oclc=1020698655}}</ref><ref name=":1">{{Cite book|last=Cox|first=David A.|url=https://www.worldcat.org/oclc/784952441|title=Galois theory|date=2012|publisher=John Wiley & Sons|isbn=978-1-118-21845-7|edition=2nd|___location=Hoboken, NJ|pages=322|oclc=784952441}}</ref> of [[Joseph-Louis Lagrange|Lagrange]] from 1771, which Galois certainly knew. It is likely that Lagrange had already been aware of the primitive element theorem for splitting fields.<ref name=":1" /> Galois then used this theorem heavily in his development of the [[Galois group]]. Since then it has been used in the development of [[Galois theory]] and the [[fundamental theorem of Galois theory]].
The primitive element theorem was proved in its modern form by [[Ernst Steinitz]], in an influential article on [[Field theory (mathematics)|field theory]] in 1910, which also contains [[Steinitz's theorem (field theory)|Steinitz's theorem]];<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|url-access=subscription}}</ref> Steinitz called the "classical" result ''Theorem of the primitive elements'' and his modern version ''Theorem of the intermediate fields''.
[[Emil Artin]] reformulated Galois theory in the 1930s without relying on primitive elements.<ref>{{cite book|last=Kleiner|first=Israel|title=A History of Abstract Algebra|date=2007|publisher=Springer|isbn=978-0-8176-4685-1|pages=64|chapter=§4.1 Galois theory|chapter-url=https://books.google.com/books?id=udj-1UuaOiIC&pg=PA64}}</ref><ref>{{Cite book|last=Artin|first=Emil|url=https://www.worldcat.org/oclc/38144376|title=Galois theory|date=1998|publisher=Dover Publications|others=Arthur N. Milgram|isbn=0-486-62342-4|edition=Republication of the 1944 revised edition of the 1942 first publication by The University Notre Dame Press|___location=Mineola, N.Y.|oclc=38144376}}</ref>
==References==
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* [http://www.mathreference.com/fld-sep,pet.html The primitive element theorem at mathreference.com]
* [http://planetmath.org/ProofOfPrimitiveElementTheorem The primitive element theorem at planetmath.org]
* [httphttps://www.math.cornell.edu/~kbrown/6310/primitive.pdf The primitive element theorem on Ken Brown's website (pdf file)]
