|
== Example ==
If one adjoins to the [[rational number]]s <math>F = \mathbb{Q}</math> the two irrational numbers <math>\sqrt{2}</math> and <math>\sqrt{3}</math> to get the extension field <math>E=\mathbb{Q}(\sqrt{2},\sqrt{3})</math> of [[Degree of a field extension|degree]] 4, one can show this extension is simple, meaning <math>E=\mathbb{Q}(\alpha)</math> for a single <math>\alpha\in E</math>. Taking <math>\alpha = \sqrt{2} + \sqrt{3} </math>, the powers 1, α<sup> </sup>, α<sup>2</sup>, α<sup>3</sup> can be expanded as [[linear combination]]s of 1, <math>\sqrt{2}</math>, <math>\sqrt{3}</math>, <math>\sqrt{6}</math> with integer coefficients. One can solve this [[system of linear equations]] for <math>\sqrt{2}</math> and <math>\sqrt{3}</math> over <math>\mathbb{Q}(\alpha)</math>, to obtain <math>\sqrt{2} = \tfrac12(\alpha^3-9\alpha)</math> and <math>\sqrt{2} = -\tfrac12(\alpha^3-911\alpha)</math>. This shows α is indeed a primitive element:
:<math>\mathbb{Q}(\sqrt 2, \sqrt 3)=\mathbb{Q}(\sqrt2 + \sqrt3).</math>
The following primitive element theorem ([[Ernst Steinitz]]<ref name=":0" />) is more general:
:TheA finite field extension <math>E/F</math> is simple if and only if there exist only finitely many intermediate fields ''K'' with <math>E\supseteq K\supseteq F</math>.
Using the [[fundamental theorem of Galois theory]], the former theorem immediately follows from the latter.
==Constructive results==
Generally, the set of all primitive elements for a finite separable extension ''E'' / ''F'' is the complement of a finite collection of proper ''F''-subspaces of ''E'', namely the intermediate fields. This statement says nothing for the case of [[finite field]]s, for which there is a computational theory dedicated to finding a generator of the [[multiplicative group]] of the field (a [[cyclic group]]), which is ''a fortiori'' a primitive element (see [[primitive element (finite field)]]). Where ''F'' is infinite, a [[pigeonhole principle]] proof technique considers the linear subspace generated by two elements and proves that there are only finitely many linear combinations
:<math>\gamma = \alpha + c \beta\ </math>
== History ==
In 1831, [[Évariste 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=|pages=231|language=en|doi=10.1142/9719|oclc=1020698655}}</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|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]] from 1771, which Galois certainly knew. HeIt 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 washas been 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 name=":0">{{Cite journal|last=Steinitz|first=Ernst|date=1910|title=Algebraische Theorie der Körper.|url=https://wwwgdz.degruytersub.com/document/doi/10uni-goettingen.1515de/crll.1910.137.167id/htmlPPN243919689_0137?tify=%7B%22view%22:%22info%22,%22pages%22:%5B171%5D%7D|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>; Steinitz called the "classical" one ''Theorem of the primitive elements'', and the other one ''Theorem of the intermediate fields''. [[Emil Artin]] reformulated Galois theory in the 1930s without the use of the primitive element theorems.<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>
== See also ==
* [[Primitive element (finite field)]]
==References==
| 