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In [[field theory (mathematics)|field theory]], the '''primitive element theorem''' is a result characterizing the [[Degree of a field extension|finite degree]] [[field extension]]s that can be generated by a single element. Such a generating element is called a '''primitive element''' of the field extension, and the extension is called a [[simple extension]] in this case. The theorem states that a finite extension is simple if and only if there are only finitely many intermediate fields. An older result, also often called "primitive element theorem", states that every finite [[separable extension|separable]] extension is simple;. it can be seen as a consequence of the formerThis theorem. These theorems implyimplies in particular that all [[Algebraic number field|algebraic number fields]] over the rational numbers, and all extensions in which both fields are finite, are simple.
== Terminology ==
:<math>\mathbb{Q}(\sqrt 2, \sqrt 3)=\mathbb{Q}(\sqrt2 + \sqrt3).</math>
== TheTheorem theoremsstatement ==
The classical primitive element theorem states:
:Every [[Separable extension|separable]] field extension of finite degree is simple.
This theorem 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.
Using the [[fundamental theorem of Galois theory]], the former theorem immediately follows from the[[Steinitz's lattertheorem (field theory)|Steinitz's theorem]]. ▼The following primitive element theorem ([[Ernst Steinitz]]<ref name=":0" />) is more general:
:A 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.
== Characteristic ''p'' ==
For a non-separable extension <math>E/F</math> of [[characteristic p]], there is nevertheless a primitive element provided the degree [''E'' : ''F''] is ''p:'' indeed, there can be no non-trivial intermediate subfields since their degrees would be factors of the prime ''p''.
When [''E'' : ''F''] = ''p''<sup>2</sup>, there may not be a primitive element (in which case there are infinitely many intermediate fields by [[Steinitz's theorem (field theory)|Steinitz's theorem]]). The simplest example is <math>E=\mathbb{F}_p(T,U)</math>, the field of rational functions in two indeterminates ''T'' and ''U'' over the [[finite field]] with ''p'' elements, and <math>F=\mathbb{F}_p(T^p,U^p)</math>. In fact, for any α = ''g''(T,U) in <math>E \setminus F</math>, the [[Frobenius endomorphism]] shows that the element ''α''<sup>''p''</sup> lies in ''F'' , so α is a root of <math>f(X)=X^p-\alpha^p\in F[X]</math>, and α cannot be a primitive element (of degree ''p''<sup>2</sup> over ''F''), but instead ''F''(α) is a non-trivial intermediate field.
Starting with a simple finite extension ''E'' = ''F''(α), let ''f'' be the [[minimal polynomial (field theory)|minimal polynomial]] of α over ''F''. If ''K'' is an intermediate subfield, then let ''g'' be the minimal polynomial of α over ''K'', and let ''L'' be the field generated over ''F'' by the coefficients of ''g''. Then since ''L'' ⊆ ''K'', the minimal polynomial of α over ''L'' must be a multiple of ''g'', so it is ''g''; this implies that the degree of ''E'' over ''L'' is the same as that over ''K'', but since ''L'' ⊆ ''K'', this means that ''L'' = ''K''. Since ''g'' is a factor of ''f'', this means that there can be no more intermediate fields than factors of ''f'', so there are only finitely many.
Going in the other direction, if ''F'' is [[finite field|finite]], then any finite extension ''E'' of ''F'' is automatically simple, so assume that ''F'' is infinite. Then ''E'' is generated over ''F'' by a finite number of elements, so it's enough to prove that ''F''(α, β) is simple for any two elements α and β in ''E''. But, considering all fields ''F''(α + ''x'' β), where ''x'' is an element of ''F'', there are only finitely many, so there must be distinct ''x<sub>0</sub>'' and ''x<sub>1</sub>'' in ''F'' for which ''F''(α + ''x<sub>0</sub>'' β) = ''F''(α + ''x<sub>1</sub>'' β). Then simple algebra shows that ''F''(α + ''x<sub>0</sub>'' β) = ''F''(α, β). For an alternative proof, observe that each of the finite number of intermediate fields is a proper linear subspace of ''E'' over ''F'', and that a finite union of proper linear subspaces of a vector space over an infinite field cannot equal the entire space. Then, taking any element in ''E'' that is not in any intermediate field, it must generate the whole of ''E'' over ''F''.<ref>Theorem 26, ''Galois Theory'', Emil Artin and Arthur N. Milgram, University of Notre Dame Press, 2nd ed., 1944.</ref><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>
==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 in 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>
with ''c'' in ''F'', that fail to generate the subfield containing both elements:
: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>.
Suppose first that <math>F</math> is infinite. If <math>F(\alpha, \beta) \supsetneq F(\alpha + c \beta)</math>, the latter field must not contain <math>\beta</math> (otherwise <math>\alpha = (\alpha + c \beta) - c\beta</math> would also be in it). Therefore, we may extend the inclusion <math>F(\alpha + c \beta) \to F(\alpha, \beta)</math> to an <math>F</math>-automorphism <math>\sigma</math> that sends <math>\beta</math> to a different root of the minimal polynomial of <math>\beta</math> over <math>F(\alpha + c \beta)</math>, since they are all different by separability (the polynomial in question is a divisor of the minimal polynomial of <math>\beta</math> over <math>f</math>). We then have
:<math>\alpha + c \beta = \sigma(\alpha + c \beta) = \sigma(\alpha) + c \sigma(\beta)</math> and therefore <math>c = \frac{\sigma(\alpha) - \alpha}{\beta - \sigma(\beta)}</math>.
Since <math>\operatorname{Gal}(F(\alpha, \beta) / F)</math> is finite (and in fact bounded by <math>[F(\alpha, \beta) : F]</math>), there are only finitely many possibilities for the value of <math>c</math>.
For the case where <math>F</math> is finite, we simply use a primitive root, which then generates the field extension.
This is almost immediate as a way of showing how Steinitz' result implies the classical result, and a bound for the number of exceptional ''c'' in terms of the number of intermediate fields results (this number being something that can be bounded itself by Galois theory and ''a priori''). Therefore, in this case trial-and-error is a possible practical method to find primitive elements.
== History ==
In his First Memoir of 1831,<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|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]]; Galois' Memoir was not published until 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=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]] 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 two primitive element theoremstheorem werewas proved in theirits 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}}</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>
==References==
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