Primitive element theorem: Difference between revisions

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{{Short description|Field theory theorem}}

In [[field theory (mathematics)|field theory]], the '''primitive element theorem''' states that every [[degree of a field extension|finite]] [[separable extension|separable]] [[field extension]] is [[Simple extension|simple]], i.e. generated by a single element. This theorem implies 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 ==

Let <math>E/F</math> be a ''[[field extension]]''. An element <math>\alpha\in E</math> is a ''primitive element'' for <math>E/F</math> if <math>E=F(\alpha),</math> i.e. if every element of <math>E</math> can be written as a [[rational function]] in <math>\alpha</math> with coefficients in <math>F</math>. If there exists such a primitive element, then <math>E/F</math> is referred to as a ''[[simple extension]]''.

If the field extension <math>E/F</math> has primitive element <math>\alpha</math> and is of finite [[Degree of a field extension|degree]] <math>n = [E:F]</math>, then every element ''x''<math>\gamma\in of ''E''</math> can be written uniquely in the form

:<math>x\gamma =f_a_0+a_1{\alpha}+\cdots+a_{n-1}{\alpha}^{n-1}+\cdots+f_1{\alpha}+f_0, </math>

wherefor unique coefficients <math>f_ia_0,a_1,\ldots,a_{n-1}\in F</math> for all ''i''. That is, the set

:<math>\{1,\alpha,\ldots,{\alpha}^{n-1}\}</math>

is a [[Basis (linear algebra)|basis]] for ''E'' as a [[vector space]] over ''F''. The degree ''n'' is equal to the degree of the [[irreducible polynomial]] of ''α'' over ''F'', the unique monic <math>f(X)\in F[X] </math> of minimal degree with ''α'' as a root (a linear dependency of <math>\{1,\alpha,\ldots,\alpha^{n-1},\alpha^n\} </math>).

If ''L'' is a [[splitting field]] of <math>f(X)</math> containing its ''n'' distinct roots <math>\alpha_1,\ldots,\alpha_n </math>, then there are ''n'' [[Homomorphism|field embeddings]] <math>\sigma_i : F(\alpha)\hookrightarrow L </math> defined by <math>\sigma_i(\alpha)=\alpha_i </math> and <math>\sigma(a)=a </math> for <math>a\in F </math>, and these extend to automorphisms of ''L'' in the [[Galois group]], <math>\sigma_1,\ldots,\sigma_n\in \mathrm{Gal}(L/F) </math>. Indeed, for an extension field with <math>[E: F]=n </math>, an element <math>\alpha</math> is a primitive element if and only if <math>\alpha</math> has ''n'' distinct conjugates <math>\sigma_1(\alpha),\ldots,\sigma_n(\alpha)</math> in some splitting field <math>L \supseteq E</math>.

== 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'', ''α''², ''α''³ 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{3} = -\tfrac12(\alpha^3-11\alpha)</math>. This shows that ''α'' is indeed a primitive element:

:<math>\mathbb{Q}(\sqrt 2, \sqrt 3)=\mathbb{Q}(\sqrt2 + \sqrt3).</math>

One may also use the following more general argument.<ref>{{Cite book |last=Lang |first=Serge |url=http://link.springer.com/10.1007/978-1-4613-0041-0 |title=Algebra |date=2002 |publisher=Springer New York |isbn=978-1-4612-6551-1 |series=Graduate Texts in Mathematics |volume=211 |___location=New York, NY |pages=243 |doi=10.1007/978-1-4613-0041-0}}</ref> The field <math>E=\Q(\sqrt 2,\sqrt 3) </math> clearly has four field automorphisms <math>\sigma_1,\sigma_2,\sigma_3,\sigma_4: E\to E </math> defined by <math>\sigma_i(\sqrt 2)=\pm\sqrt 2 </math> and <math>\sigma_i(\sqrt 3)=\pm\sqrt 3 </math> for each choice of signs. The minimal polynomial <math>f(X)\in\Q[X] </math> of <math>\alpha=\sqrt 2+\sqrt 3 </math> must have <math>f(\sigma_i(\alpha)) = \sigma_i(f(\alpha)) = 0 </math>, so <math>f(X) </math> must have at least four distinct roots <math>\sigma_i(\alpha)=\pm\sqrt 2 \pm\sqrt 3 </math>. Thus <math>f(X) </math> has degree at least four, and <math>[\Q(\alpha):\Q]\geq 4 </math>, but this is the degree of the entire field, <math>[E:\Q]=4 </math>, so <math>E = \Q(\alpha ) </math>.

== Theorem statement ==

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For a non-separable extension <math>E/F</math> of [[characteristic p]], there is nevertheless a primitive element provided the degree [''E''&nbsp;:&nbsp;''F''] is ''p:'' indeed, there can be no non-trivial intermediate subfields since their degrees would be factors of the prime ''p''.

When [''E''&nbsp;:&nbsp;''F''] = ''p''², 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 &<math>\alpha; = ''g''(T,U) </math> in <math>E \setminus F</math>, the [[Frobenius endomorphism]] shows that the element ''&alpha;''<supmath>''\alpha^p''</supmath> lies in ''F'' , so ''&alpha;'' is a root of <math>f(X)=X^p-\alpha^p\in F[X]</math>, and ''&alpha;'' cannot be a primitive element (of degree ''p''² over ''F''), but instead ''F''(''&alpha;'') is a non-trivial intermediate field.

== Proof ==

Suppose first that <math>F</math> is infinite. IfBy induction, it suffices to prove that any finite extension <math>E=F(\alphabeta, \betagamma) </math> is simple. For <math>c\supsetneqin F(</math>, suppose <math>\alpha = \beta+ c \beta)gamma </math>, thefails latterto fieldbe musta notprimitive containelement, <math>F(\alpha)\subsetneq F(\beta,\gamma)</math>. (otherwiseThen <math>\alpha =gamma\notin F(\alpha)</math>, +since cotherwise <math>\beta) -= \alpha-c\gamma\in F(\alpha)=F(\beta,\gamma)</math>. wouldConsider alsothe beminimal inpolynomials it).of Therefore<math>\beta,\gamma we may</math> extend the inclusionover <math>F(\alpha) + c</math>, \betarespectively <math>f(X), g(X) \toin F(\alpha, \beta)[X]</math>, toand antake a splitting field <math>FL </math>-automorphism containing all roots <math>\sigmabeta,\beta',\ldots</math> that sendsof <math>\betaf(X) </math> toand a<math>\gamma,\gamma',\ldots different root of the minimal polynomial</math> of <math>\betag(X) </math>. overSince <math>\gamma\notin F(\alpha + c \beta)</math>, sincethere theyis areanother allroot different<math>\gamma'\neq by\gamma</math>, separability (the polynomial in question isand a divisorfield ofautomorphism the<math>\sigma:L\to minimalL polynomial</math> ofwhich fixes <math>F(\betaalpha) </math> overand takes <math>f\sigma(\gamma)=\gamma' </math>). (PossiblyWe wethen have to enlarge <math>F\sigma(\alpha,) =\beta)alpha </math>.), We then haveand:

:<math>\alphabeta + c \betagamma = \sigma(\alphabeta + c \betagamma) = \sigma(\alphabeta) + c \, \sigma(\betagamma) </math>, and therefore <math>c = \frac{\sigma(\alphabeta) - \alphabeta}{\betagamma - \sigma(\betagamma)}</math>.

Since there are only finitely many possibilities for <math>\sigma(\beta)=\beta'</math>, thereand are<math>\sigma(\gamma)=\gamma'</math>, only finitely many possibilities<math>c\in forF</math> thefail valueto ofgive a primitive element <math>\alpha=\beta+c\gamma</math>. All other values give <math>F(\alpha)=F(\beta,\gamma) </math>.

For the case where <math>F</math> is finite, we simply usetake <math>\alpha</math> to be a primitive[[Primitive root, whichmodulo thenn|primitive generatesroot]] of the fieldfinite extension field <math>E </math>.

== 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" oneresult ''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)]

[[Category:Field (mathematics)]]