Primitive element theorem: Difference between revisions

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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]] 4 <math>n = [E:F]</math>, then every element ''xγ'' of ''E'' 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 ''f''(''x'') 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>.

== 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 irreducible 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) \supsetneq F(\alpha + c \beta)</math> foris simple. someFor <math>c\in F</math>, the latter field must not contain <math>\beta</math> (otherwisesuppose <math>\alpha = (\alpha beta+ c \beta)gamma - c\beta</math> wouldfails alsoto be ina it).primitve Thereforeelement, we may extend the inclusion <math>F(\alpha + c \beta) \tosubsetneq F(\alphabeta, \betagamma)</math>. to anThen <math>\gamma\notin F(\alpha)</math>-automorphism, since otherwise <math>\sigma</math>beta that= sends\alpha-c\gamma\in <math>F(\alpha)=F(\beta,\gamma)</math>. to a different root ofConsider the minimal polynomialpolynomials of <math>\beta,\gamma </math> over <math>F(\alpha) + c \beta)</math>, sincerespectively they are all different by separability<math>f(X), g(the polynomialX) \in questionF(\alpha)[X]</math>, isand take a divisorsplitting offield the<math>L minimal</math> polynomialcontaining ofall roots <math>\beta,\beta',\ldots</math> over of <math>f(X) </math>). (Weand have<math>\gamma,\gamma',\ldots to</math> enlargeof <math>Fg(X) </math>. Since <math>\alpha,gamma\notin F(\betaalpha)</math>, tothere ais splittinganother field ifroot it isn<math>\gamma't one\neq already\gamma</math>, and ina thisfield caseautomorphism extend<math>\sigma:L\to theL inclusion</math> towhich thisfixes larger<math>F(\alpha) field.</math> and takes <math>\sigma(\gamma)=\gamma' </math>. We then have <math>\sigma(\alpha) =\alpha </math>, and:

:<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(\alphabeta)=\beta'</math> and <math>\sigma(\betagamma)=\gamma'</math>, there are only finitely many possibilities for the value of <math>c\in F</math>. (Counting the possibilities even leadsfail to angive uppera bound.)primitive For all other values ofelement <math>\alpha=\beta+c\in Fgamma</math>. thenAll other values give <math>F(\alpha,\beta) = F(\alpha+cbeta,\betagamma) </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|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}}</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==