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== History ==
In 1831, [[Évariste Galois|Evariste Galois]] sketched a proof of the classical primitive element theorem in his First Memoir<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|others=|isbn=978-3-03719-104-0|___location=Zürich|oclc=757486602}}</ref>, 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|url=https://www.worldscientific.com/worldscibooks/10.1142/9719|title=Galois' Theory of Algebraic Equations|date=2016-02-XX|publisher=WORLD SCIENTIFIC|isbn=978-981-4704-69-4|edition=2|___location=p. 231|language=en|doi=10.1142/9719}}</ref> (as remarked by the referee [[Siméon Denis Poisson]]; Galois' Memoir was only published in 1846) 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=2016-02-XX|publisher=WORLD SCIENTIFIC|isbn=978-981-4704-69-4|edition=2|___location=p. 135|language=en|doi=10.1142/9719}}</ref><ref>{{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=p. 322|pages=|oclc=}}</ref> of [[Joseph-Louis Lagrange]] from 1771, which Galois certainly knew. He then used this theorem heavily in his development of the [[Galois group]]. Since then it was used in the development of [[Galois theory]] and the [[fundamental theorem of Galois theory]]. The two primitive element theorems were proved in their modern form by [[Ernst Steinitz]], in an influential article on [[Field theory (mathematics)|field theory]] in 1910.<ref>{{Cite journal|last=Steinitz|first=Ernst|date=1910|title=Algebraische Theorie der Körper.|url=https://www.degruyter.com/document/doi/10.1515/crll.1910.137.167/html|journal=Journal für die reine und angewandte Mathematik|language=de|volume=137|pages=167–309|doi=10.1515/crll.1910.137.167|issn=1435-5345}}</ref> [[Emil Artin]] reformulated Galois theory in the 1930s without the use of the primitive element theorems.<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>
== See also ==
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