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Algebraic extension
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{{Short description|Extension of a mathematical field with polynomial roots}} {{use mdy dates|date=September 2021}} {{Use American English|date = January 2019}} In [[mathematics]], an '''algebraic extension''' is a [[field extension]] {{math|''L''/''K''}} such that every element of the larger [[field (mathematics)|field]] {{mvar|L}} is [[algebraic element|algebraic]] over the smaller field {{mvar|K}}; that is, every element of {{mvar|L}} is a root of a non-zero [[polynomial]] with coefficients in {{mvar|K}}.<ref>Fraleigh (2014), Definition 31.1, p. 283.</ref><ref>Malik, Mordeson, Sen (1997), Definition 21.1.23, p. 453.</ref> A field extension that is not algebraic, is said to be [[Field extension#Transcendental extension|transcendental]], and must contain [[transcendental element]]s, that is, elements that are not algebraic.<ref>Fraleigh (2014), Definition 29.6, p. 267.</ref><ref>Malik, Mordeson, Sen (1997), Theorem 21.1.8, p. 447.</ref> The algebraic extensions of the field <math>\Q</math> of the [[rational number]]s are called [[algebraic number field]]s and are the main objects of study of [[algebraic number theory]]. Another example of a common algebraic extension is the extension <math>\Complex/\R</math> of the [[real number]]s by the [[complex number]]s. ==Some properties== All transcendental extensions are of infinite [[degree of a field extension|degree]]. This in turn implies that all finite extensions are algebraic.<ref>See also Hazewinkel et al. (2004), p. 3.</ref> The [[converse (logic)|converse]] is not true however: there are infinite extensions which are algebraic.<ref>Fraleigh (2014), Theorem 31.18, p. 288.</ref> For instance, the field of all [[algebraic number]]s is an infinite algebraic extension of the rational numbers.<ref>Fraleigh (2014), Corollary 31.13, p. 287.</ref> Let {{math|''E''}} be an extension field of {{math|''K''}}, and {{math|''a'' ∈ ''E''}}. The smallest subfield of {{math|''E''}} that contains {{math|''K''}} and {{mvar|a}} is commonly denoted <math>K(a).</math> If {{mvar|''a''}} is algebraic over {{math|''K''}}, then the elements of {{math|''K''(''a'')}} can be expressed as polynomials in {{mvar|''a''}} with coefficients in ''K''; that is, <math>K(a)=K[a]</math>, the smallest [[ring (mathematics)|ring]] containing {{math|''K''}} and {{mvar|a}}. In this case, <math>K(a)</math> is a finite extension of {{mvar|K}} and all its elements are algebraic over {{mvar|K}}. In particular, <math>K(a)</math> is a {{mvar|K}}-vector space with basis <math>\{1,a,...,a^{d-1}\}</math>, where ''d'' is the degree of the [[Minimal polynomial (field theory)|minimal polynomial]] of {{mvar|a}}.<ref>Fraleigh (2014), Theorem 30.23, p. 280.</ref> These properties do not hold if {{mvar|a}} is not algebraic. For example, <math>\Q(\pi)\neq \Q[\pi],</math> and they are both infinite dimensional vector spaces over <math>\Q.</math><ref>Fraleigh (2014), Example 29.8, p. 268.</ref> An [[algebraically closed field]] ''F'' has no proper algebraic extensions, that is, no algebraic extensions ''E'' with ''F'' < ''E''.<ref>Fraleigh (2014), Corollary 31.16, p. 287.</ref> An example is the field of complex numbers. Every field has an algebraic extension which is algebraically closed (called its [[algebraic closure]]), but [[mathematical proof|proving]] this in general requires some form of the [[axiom of choice]].<ref>Fraleigh (2014), Theorem 31.22, p. 290.</ref> An extension ''L''/''K'' is algebraic [[if and only if]] every sub ''K''-[[Algebra over a field|algebra]] of ''L'' is a field. ==Properties== The following three properties hold:<ref>Lang (2002) p.228</ref> # If ''E'' is an algebraic extension of ''F'' and ''F'' is an algebraic extension of ''K'' then ''E'' is an algebraic extension of ''K''. # If ''E'' and ''F'' are algebraic extensions of ''K'' in a common overfield ''C'', then the [[compositum]] ''EF'' is an algebraic extension of ''K''. # If ''E'' is an algebraic extension of ''F'' and ''E'' > ''K'' > ''F'' then ''E'' is an algebraic extension of ''K''. These finitary results can be generalized using transfinite induction: {{ordered list|start=4 | The [[union (set theory)|union]] of any chain of algebraic extensions over a base field is itself an algebraic extension over the same base field. }} This fact, together with [[Zorn's lemma]] (applied to an appropriately chosen [[poset]]), establishes the existence of [[algebraic closure]]s. ==Generalizations== {{Main|Substructure (mathematics)}} [[Model theory]] generalizes the notion of algebraic extension to arbitrary theories: an [[embedding]] of ''M'' into ''N'' is called an '''algebraic extension''' if for every ''x'' in ''N'' there is a [[Well-formed formula|formula]] ''p'' with parameters in ''M'', such that ''p''(''x'') is true and the set :<math>\left\{y\in N \mid p(y)\right\}</math> is finite. It turns out that applying this definition to the theory of fields gives the usual definition of algebraic extension. The [[Galois group]] of ''N'' over ''M'' can again be defined as the [[Group (mathematics)|group]] of [[automorphism]]s, and it turns out that most of the theory of Galois groups can be developed for the general case. == Relative algebraic closures == Given a field ''k'' and a field ''K'' containing ''k'', one defines the '''relative algebraic closure''' of ''k'' in ''K'' to be the subfield of ''K'' consisting of all elements of ''K'' that are algebraic over ''k'', that is all elements of ''K'' that are a root of some nonzero polynomial with coefficients in ''k''. == See also == * [[Integral element]] * [[Lüroth's theorem]] * [[Galois extension]] * [[Separable extension]] * [[Normal extension]] == Notes == <references/> ==References== * {{citation|first=John B.|last=Fraleigh|title=A First Course in Abstract Algebra|year=2014|publisher=Pearson|isbn=978-1-292-02496-7}} * {{citation |first1=Michiel |last1=Hazewinkel |author-link1=Michiel Hazewinkel |first2=Nadiya |last2=Gubareni |first3=Nadezhda Mikhaĭlovna |last3=Gubareni |first4=Vladimir V. |last4=Kirichenko |title=Algebras, rings and modules |url=https://books.google.com/books?id=AibpdVNkFDYC |volume=1 |year=2004 |publisher=Springer |isbn=1-4020-2690-0}} * {{Lang Algebra|edition=3|chapter=V.1:Algebraic Extensions|pages=223ff}} * {{citation|first1=D. B.|last1=Malik|first2=John N.|last2=Mordeson|first3=M. K.|last3=Sen|title=Fundamentals of Abstract Algebra|year=1997|publisher=McGraw-Hill|isbn=0-07-040035-0}} * {{citation |last=McCarthy |first=Paul J. |title=Algebraic extensions of fields |url=https://books.google.com/books?id=p-8WAgAAQBAJ |zbl=0768.12001 |location=New York |publisher=Dover Publications |year=1991 |orig-year=corrected reprint of 2nd edition, 1976 |isbn=0-486-66651-4}} * {{citation |first=Steven |last=Roman |title=Field Theory |url=https://archive.org/details/springer_10.1007-978-1-4612-2516-4 |series=GTM 158 |year=1995 |publisher=Springer-Verlag |isbn=9780387944081}} * {{citation |first=Joseph J. |last=Rotman |title=Advanced Modern Algebra |url=https://archive.org/details/advancedmodernal0000rotm |year=2002 |publisher=Prentice Hall |isbn=9780130878687}} {{DEFAULTSORT:Algebraic Extension}} [[Category:Field extensions]]
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