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| In [[abstract algebra]], '''field extensions''' are the main object of study in [[field theory (mathematics)|field theory]]. The general idea is to start with a base [[field (mathematics)|field]] and construct in some manner a larger field that contains the base field and satisfies additional properties. For instance, the set '''Q'''(√2) = {''a'' + ''b''√2 | ''a'', ''b'' ∈ '''Q'''} is the smallest extension of '''Q''' which includes every real solution to the equation ''x''<sup>2</sup> = 2.
| | I'm Nannette and I live with my husband and our three children in Park Avenue, in the QLD south area. My hobbies are Roller Derby, Kiteboarding and Rock stacking.<br><br>my web blog - [http://hetilainat.fi hetilainat] |
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| == Definitions ==
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| Let ''L'' be a [[Field (mathematics)|field]]. A '''subfield''' of ''L'' is a [[subset]] ''K'' of ''L'' which is [[Closure (mathematics)|closed]] under the field operations of ''L'' and under taking inverses in ''L''. In other words, ''K'' is a field with respect to the field operations inherited from ''L''. The larger field ''L'' is then said to be an '''extension field''' of ''K''. To simplify notation and terminology, one says that ''L'' / ''K'' (read as "''L'' over ''K''") is a '''field extension''' to signify that ''L'' is an extension field of ''K''.
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| If ''L'' is an extension of ''F'' which is in turn an extension of ''K'', then ''F'' is said to be an '''intermediate field''' (or '''intermediate extension''' or '''subextension''') of the field extension ''L'' /''K''.
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| Given a field extension ''L'' /''K'' and a subset ''S'' of ''L'', the smallest subfield of ''L'' which contains ''K'' and ''S'' is denoted by ''K''(''S'')—i.e. ''K''(''S'') is the field generated by '''[[adjunction (field theory)|adjoining]]''' the elements of ''S'' to ''K''. If ''S'' consists of only one element ''s'', ''K''(''s'') is a shorthand for ''K''({''s''}). A field extension of the form ''L'' = ''K''(''s'') is called a [[simple extension]] and ''s'' is called a [[primitive element (field theory)|primitive element]] of the extension.
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| Given a field extension ''L'' /''K'', the larger field ''L'' can be considered as a [[vector space]] over ''K''. The elements of ''L'' are the "vectors" and the elements of ''K'' are the "scalars", with vector addition and scalar multiplication obtained from the corresponding field operations. The [[dimension (vector space)|dimension]] of this vector space is called the [[degree of a field extension|'''degree''' of the extension]] and is denoted by [''L'' : ''K''].
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| An extension of degree 1 (that is, one where ''L'' is equal to ''K'') is called a '''trivial extension'''. Extensions of degree 2 and 3 are called '''quadratic extensions''' and '''cubic extensions''', respectively. Depending on whether the degree is finite or infinite the extension is called a '''finite extension''' or '''infinite extension'''.
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| == Caveats ==
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| The notation ''L'' /''K'' is purely formal and does not imply the formation of a [[quotient ring]] or [[quotient group]] or any other kind of division. Instead the slash expresses the word "over". In some literature the notation ''L'':''K'' is used.
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| It is often desirable to talk about field extensions in situations where the small field is not actually contained in the larger one, but is naturally embedded. For this purpose, one abstractly defines a field extension as an [[injective function|injective]] [[ring homomorphism]] between two fields.
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| ''Every'' non-zero ring homomorphism between fields is injective because fields do not possess nontrivial proper ideals, so field extensions are precisely the [[morphism]]s in the [[category of fields]].
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| Henceforth, we will suppress the injective homomorphism and assume that we are dealing with actual subfields.
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| == Examples ==
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| The field of [[complex number]]s '''C''' is an extension field of the field of [[real number]]s '''R''', and '''R''' in turn is an extension field of the field of [[rational number]]s '''Q'''. Clearly then, '''C'''/'''Q''' is also a field extension. We have ['''C''' : '''R'''] = 2 because {1,''i''} is a basis, so the extension '''C'''/'''R''' is finite. This is a simple extension because '''C'''='''R'''(<math>i</math>). ['''R''' : '''Q'''] = <math>\mathfrak c</math> (the [[cardinality of the continuum]]), so this extension is infinite.
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| The set '''Q'''(√2) = {''a'' + ''b''√2 | ''a'', ''b'' ∈ '''Q'''} is an extension field of '''Q''', also clearly a simple extension. The degree is 2 because {1, √2} can serve as a basis. Finite extensions of '''Q''' are also called [[algebraic number field]]s and are important in [[number theory]].
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| Another extension field of the rationals, quite different in flavor, is the field of [[p-adic number]]s '''Q'''<sub>''p''</sub> for a prime number ''p''.
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| It is common to construct an extension field of a given field ''K'' as a [[quotient ring]] of the [[polynomial ring]] ''K''[''X''] in order to "create" a [[root of a function|root]] for a given polynomial ''f''(''X''). Suppose for instance that ''K'' does not contain any element ''x'' with ''x''<sup>2</sup> = −1. Then the polynomial ''X''<sup>2</sup> + 1 is [[irreducible polynomial|irreducible]] in ''K''[''X''], consequently the [[ideal (ring theory)|ideal]] (''X''<sup>2</sup> + 1) generated by this polynomial is [[maximal ideal|maximal]], and ''L'' = ''K''[''X'']/(''X''<sup>2</sup> + 1) is an extension field of ''K'' which ''does'' contain an element whose square is −1 (namely the residue class of ''X'').
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| By iterating the above construction, one can construct a [[splitting field]] of any polynomial from ''K''[''X'']. This is an extension field ''L'' of ''K'' in which the given polynomial splits into a product of linear factors.
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| If ''p'' is any [[prime number]] and ''n'' is a positive integer, we have a [[finite field]] GF(''p''<sup>''n''</sup>) with ''p''<sup>''n''</sup> elements; this is an extension field of the finite field GF(''p'') = '''Z'''/''p'''''Z''' with ''p'' elements.
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| Given a field ''K'', we can consider the field ''K''(''X'') of all [[rational function]]s in the variable ''X'' with coefficients in ''K''; the elements of ''K''(''X'') are fractions of two [[polynomial]]s over ''K'', and indeed ''K''(''X'') is the [[field of fractions]] of the polynomial ring ''K''[''X'']. This field of rational functions is an extension field of ''K''. This extension is infinite.
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| Given a [[Riemann surface]] ''M'', the set of all [[meromorphic function]]s defined on ''M'' is a field, denoted by '''C'''(''M''). It is an extension field of '''C''', if we identify every complex number with the corresponding [[constant function]] defined on ''M''.
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| Given an [[algebraic variety]] ''V'' over some field ''K'', then the [[function field of an algebraic variety|function field]] of ''V'', consisting of the rational functions defined on ''V'' and denoted by ''K''(''V''), is an extension field of ''K''.
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| == Elementary properties ==
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| If ''L''/''K'' is a field extension, then ''L'' and ''K'' share the same 0 and the same 1. The additive group (''K'',+) is a [[subgroup]] of (''L'',+), and the multiplicative group (''K''−{0},·) is a subgroup of (''L''−{0},·). In particular, if ''x'' is an element of ''K'', then its additive inverse −''x'' computed in ''K'' is the same as the additive inverse of ''x'' computed in ''L''; the same is true for multiplicative inverses of non-zero elements of ''K''.
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| In particular then, the [[characteristic (algebra)|characteristics]] of ''L'' and ''K'' are the same.
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| == Algebraic and transcendental elements and extensions ==
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| If ''L'' is an extension of ''K'', then an element of ''L'' which is a [[root of a function|root]] of a nonzero [[polynomial]] over ''K'' is said to be [[algebraic element|algebraic]] over ''K''. Elements that are not algebraic are called [[transcendental element|transcendental]]. For example:
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| * In '''C'''/'''R''', ''i'' is algebraic because it is a root of ''x''<sup>2</sup> + 1.
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| * In '''R'''/'''Q''', √2 + √3 is algebraic, because it is a root<ref>{{cite web|url=http://www.wolframalpha.com/input/?i=sqrt(2)%2Bsqrt(3)|title=Wolfram{{!}}Alpha input: ''sqrt(2)+sqrt(3)''|accessdate=2010-06-14}}</ref> of ''x''<sup>4</sup> − 10''x''<sup>2</sup> + 1
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| * In '''R'''/'''Q''', ''e'' is transcendental because there is no polynomial with rational coefficients that has ''e'' as a root (see [[transcendental number]])
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| * In '''C'''/'''R''', ''e'' is algebraic because it is the root of ''x'' − ''e''
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| The special case of '''C'''/'''Q''' is especially important, and the names [[algebraic number]] and transcendental number are used to describe the complex numbers that are algebraic and transcendental (respectively) over '''Q'''.
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| If every element of ''L'' is algebraic over ''K'', then the extension ''L''/''K'' is said to be an ''[[algebraic extension]]''; otherwise it is said to be a ''transcendental extension''.
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| A subset ''S'' of ''L'' is called [[algebraically independent]] over ''K'' if no non-trivial polynomial relation with coefficients in ''K'' exists among the elements of ''S''. The largest cardinality of an algebraically independent set is called the [[transcendence degree]] of ''L''/''K''. It is always possible to find a set ''S'', algebraically independent over ''K'', such that ''L''/''K''(''S'') is algebraic. Such a set ''S'' is called a [[transcendence basis]] of ''L''/''K''. All transcendence bases have the same cardinality, equal to the transcendence degree of the extension. An extension ''L''/''K'' is said to be ''purely transcendental'' if and only if there exists a transcendence basis ''S'' of ''L''/''K'' such that ''L''=''K''(''S''). Such an extension has the property that all elements of ''L'' except those of ''K'' are transcendental over ''K'', but, however, there are extensions with this property which are not purely transcendental. In addition, if ''L''/''K'' is purely transcendental and ''S'' is a transcendence basis of the extension, it doesn't necessarily follow that ''L''=''K''(''S''). (For example, consider the extension '''Q'''(''x'',√''x'')/'''Q''', where ''x'' is transcendental over '''Q'''. The set {''x''} is algebraically independent since ''x'' is transcendental. Obviously, the extension '''Q'''(''x'',√''x'')/'''Q'''(''x'') is algebraic, hence {''x''} is a transcendence basis. It doesn't generate the whole extension because there is no polynomial expression in ''x'' for √''x''. But it is easy to see that {√''x''} is a transcendence basis that generates '''Q'''(''x'',√''x'')), so this extension is indeed purely transcendental.)
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| It can be shown that an extension is algebraic [[if and only if]] it is the
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| union of its finite subextensions. In particular, every finite extension is algebraic. For example,
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| * '''C'''/'''R''' and '''Q'''(√2)/'''Q''', being finite, are algebraic.
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| * '''R'''/'''Q''' is transcendental, although not purely transcendental.
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| * ''K''(''X'')/''K'' is purely transcendental.
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| A simple extension is finite if generated by an algebraic element, and purely transcendental if generated by a transcendental element. So
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| * '''R'''/'''Q''' is not simple, as it is neither finite nor purely transcendental.
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| Every field ''K'' has an [[algebraic closure]]; this is essentially the largest extension field of ''K'' which is algebraic over ''K'' and which contains all roots of all polynomial equations with coefficients in ''K''. For example, '''C''' is the algebraic closure of '''R'''.
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| == Normal, separable and Galois extensions ==
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| An algebraic extension ''L''/''K'' is called [[normal extension|normal]] if every [[irreducible polynomial]] in ''K''[''X''] that has a root in ''L'' completely factors into linear factors over ''L''. Every algebraic extension ''F''/''K'' admits a normal closure ''L'', which is an extension field of ''F'' such that ''L''/''K'' is normal and which is minimal with this property.
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| An algebraic extension ''L''/''K'' is called [[separable extension|separable]] if the [[Minimal polynomial (field theory)|minimal polynomial]] of every element of ''L'' over ''K'' is [[separable polynomial|separable]], i.e., has no repeated roots in an [[algebraic closure]] over ''K''. A [[Galois extension]] is a field extension that is both normal and separable.
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| A consequence of the [[primitive element theorem]] states that every finite separable extension has a primitive element (i.e. is simple).
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| Given any field extension ''L''/''K'', we can consider its '''automorphism group''' Aut(''L''/''K''), consisting of all field [[automorphism]]s ''α'': ''L'' → ''L'' with ''α''(''x'') = ''x'' for all ''x'' in ''K''. When the extension is Galois this automorphism group is called the [[Galois group]] of the extension. Extensions whose Galois group is [[abelian group|abelian]] are called [[abelian extension]]s.
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| For a given field extension ''L''/''K'', one is often interested in the intermediate fields ''F'' (subfields of ''L'' that contain ''K''). The significance of Galois extensions and Galois groups is that they allow a complete description of the intermediate fields: there is a [[bijection]] between the intermediate fields and the [[subgroup]]s of the Galois group, described by the [[fundamental theorem of Galois theory]].
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| == Generalizations ==
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| Field extensions can be generalized to [[ring extensions]] which consist of a [[ring (mathematics)|ring]] and one of its [[subring]]s. A closer non-commutative analog are [[central simple algebra]]s (CSAs) – ring extensions over a field, which are [[simple algebra]] (no non-trivial 2-sided ideals, just as for a field) and where the center of the ring is exactly the field. For example, the only finite field extension of the real numbers is the complex numbers, while the quaternions are a central simple algebra over the reals, and all CSAs over the reals are [[Brauer equivalent]] to the reals or the quaternions. CSAs can be further generalized to [[Azumaya algebra]]s, where the base field is replaced by a commutative [[local ring]].
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| == Extension of scalars ==
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| {{main|Extension of scalars}}
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| Given a field extension, one can "[[Extension of scalars|extend scalars]]" on associated algebraic objects. For example, given a real vector space, one can produce a complex vector space via [[complexification]]. In addition to vector spaces, one can perform extension of scalars for [[associative algebra]]s over defined over the field, such as polynomials or [[group algebra]]s and the associated [[group representation]]s. Extension of scalars of polynomials is often used implicitly, by just considering the coefficients as being elements of a larger field, but may also be considered more formally. Extension of scalars has numerous applications, as discussed in [[extension of scalars#Applications|extension of scalars: applications]].
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| == See also ==
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| * [[Field theory (mathematics)|Field theory]]
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| * [[Glossary of field theory]]
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| * [[Tower of fields]]
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| * [[Primary extension]]
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| * [[Regular extension]]
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| == Notes ==
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| <references/>
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| ==References==
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| *{{Lang Algebra | edition=3r2004}}
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| ==External links==
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| * {{springer|title=Extension of a field|id=p/e036970}}
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| [[Category:Field extensions]]
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