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:Every [[Separable extension|separable]] field extension of finite degree is simple.
This theorem, by [[Ernst Steinitz]] (1910), applies to [[algebraic number field]]s, i.e. finite extensions of the rational numbers '''Q''', since '''Q''' has [[characteristic (algebra)|characteristic]] 0 and therefore every finite extension over '''Q''' is separable.
The following primitive element theorem is more general:
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:as <math>F(\alpha,\beta)/F(\alpha+c\beta)</math> is a separable extension, if <math>F(\alpha+c\beta) \subsetneq F(\alpha,\beta)</math> there exists a non-trivial embedding <math>\sigma : F(\alpha,\beta)\to \overline{F}</math> whose restriction to <math>F(\alpha+c\beta)</math> is the identity which means <math> \sigma(\alpha)+c \sigma(\beta) = \alpha+c \beta</math> and <math>\sigma(\beta) \ne \beta</math> so that <math> c = \frac{\sigma(\alpha)-\alpha}{\beta-\sigma(\beta)}</math>. This expression for ''c'' can take only <math>[F(\alpha):F] [F(\beta):F]</math> different values. For all other value of <math>c\in F</math> then <math>F(\alpha,\beta) = F(\alpha+c\beta)</math>.
This is almost immediate as a way of showing how
== 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>{{
== See also ==
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