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Field extension

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Construction of a larger algebraic field by "adding elements" to a smaller field

Inmathematics, particularly inalgebra, afield extension is a pair offields $K\subseteq L$ , such that the operations ofK are those ofLrestricted toK. In this case,L is anextension field ofK andK is asubfield ofL.^[1]^[2]^[3] For example, under the usual notions ofaddition andmultiplication, thecomplex numbers are an extension field of thereal numbers; the real numbers are a subfield of the complex numbers.

Field extensions are fundamental inalgebraic number theory, and in the study ofpolynomial roots throughGalois theory, and are widely used inalgebraic geometry.

Subfield

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Asubfield $K {\displaystyle K}$ of afield $L {\displaystyle L}$ is asubset $K\subseteq L$ that is a field with respect to the field operations inherited from $L {\displaystyle L}$ . Equivalently, a subfield is a subset that contains themultiplicative identity $1 {\displaystyle 1}$ , and isclosed under the operations of addition, subtraction, multiplication, and taking theinverse of a nonzero element of $K {\displaystyle K}$ .

As1 – 1 = 0, the latter definition implies $K {\displaystyle K}$ and $L {\displaystyle L}$ have the samezero element.

For example, the field ofrational numbers is a subfield of thereal numbers, which is itself a subfield of the complex numbers. More generally, the field of rational numbers is (or isisomorphic to) a subfield of any field ofcharacteristic $0 {\displaystyle 0}$ .

Thecharacteristic of a subfield is the same as the characteristic of the larger field.

Extension field

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If $K {\displaystyle K}$ is a subfield of $L {\displaystyle L}$ , then $L {\displaystyle L}$ is anextension field or simplyextension of $K {\displaystyle K}$ , and this pair of fields is afield extension. Such a field extension is denoted $L/K$ (read as " $L {\displaystyle L}$ over $K {\displaystyle K}$ ").

If $L {\displaystyle L}$ is an extension of $F {\displaystyle F}$ , which is in turn an extension of $K {\displaystyle K}$ , then $F {\displaystyle F}$ is said to be anintermediate field (orintermediate extension orsubextension) of $L/K$ .

Given a field extension $L/K$ , the larger field $L {\displaystyle L}$ is a $K {\displaystyle K}$ -vector space. Thedimension of this vector space is called thedegree of the extension and is denoted by $[L:K]$ .

The degree of an extension is 1 if and only if the two fields are equal. In this case, the extension is atrivial extension. Extensions of degree 2 and 3 are calledquadratic extensions andcubic extensions, respectively. Afinite extension is an extension that has a finite degree.

Given two extensions $L/K$ and $M/L$ , the extension $M/K$ is finite if and only if both $L/K$ and $M/L$ are finite. In this case, one has

[M:K]=[M:L]\cdot [L:K].

Given a field extension $L/K$ and a subset $S {\displaystyle S}$ of $L {\displaystyle L}$ , there is a smallest subfield of $L {\displaystyle L}$ that contains $K {\displaystyle K}$ and $S {\displaystyle S}$ . It is the intersection of all subfields of $L {\displaystyle L}$ that contain $K {\displaystyle K}$ and $S {\displaystyle S}$ , and is denoted by $K(S)$ (read as " $K {\displaystyle K}$ adjoin $S {\displaystyle S}$ "). One says that $K(S)$ is the fieldgenerated by $S {\displaystyle S}$ over $K {\displaystyle K}$ , and that $S {\displaystyle S}$ is agenerating set of $K(S)$ over $K {\displaystyle K}$ . When $S=\{x_{1},\ldots ,x_{n}\}$ is finite, one writes $K(x_{1},\ldots ,x_{n})$ instead of $K(\{x_{1},\ldots ,x_{n}\}),$ and one says that $K(S)$ isfinitely generated over $K {\displaystyle K}$ . If $S {\displaystyle S}$ consists of a single element $s {\displaystyle s}$ , the extension $K(s)/K$ is called asimple extension^[4]^[5] and $s {\displaystyle s}$ is called aprimitive element of the extension.^[6]

An extension field of the form $K(S)$ is often said to result from theadjunction of $S {\displaystyle S}$ to $K {\displaystyle K}$ .^[7]^[8]

Incharacteristic 0, every finite extension is a simple extension. This is theprimitive element theorem, which does not hold true for fields of non-zero characteristic.

If a simple extension $K(s)/K$ is not finite, the field $K(s)$ is isomorphic to the field ofrational fractions in $s {\displaystyle s}$ over $K {\displaystyle K}$ .

Caveats

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The notationL /K is purely formal and does not imply the formation of aquotient ring orquotient group or any other kind of division. Instead the slash expresses the word "over". Some authors use the notationsL :K orL |K, while others may simply indicate verbally that $L\supset K$ is a field extension. Towers of extensions are often depicted diagrammatically. For example, the diagram below depicts the situation whereL is an extension ofK andK is an extension ofF:

{\begin{array}{c}L\\{\Big |}\\K\\{\Big |}\\F\end{array}}

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 aninjective ring homomorphism between two fields.Every ring homomorphism between fields is injective because fields do not possess nontrivial properideals, so field extensions are precisely themorphisms in thecategory of fields.

Henceforth, we will suppress the injective homomorphism and assume that we are dealing with actual subfields.

Examples

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The field of complex numbers $\mathbb {C}$ is an extension field of the field ofreal numbers $\mathbb {R}$ , and $\mathbb {R}$ in turn is an extension field of the field of rational numbers $\mathbb {Q}$ . Clearly then, $\mathbb {C} /\mathbb {Q}$ is also a field extension. We have $[\mathbb {C} :\mathbb {R} ]=2$ because $\{1,i\}$ is a basis, so the extension $\mathbb {C} /\mathbb {R}$ is finite. This is a simple extension because $\mathbb {C} =\mathbb {R} (i).$ $[\mathbb {R} :\mathbb {Q} ]={\mathfrak {c}}$ (thecardinality of the continuum), so this extension is infinite.

The field

\mathbb {Q} ({\sqrt {2}})=\left\{a+b{\sqrt {2}}\mid a,b\in \mathbb {Q} \right\},

is an extension field of $\mathbb {Q} ,$ also clearly a simple extension. The degree is 2 because $\left\{1,{\sqrt {2}}\right\}$ can serve as a basis.

The field

{\begin{aligned}\mathbb {Q} \left({\sqrt {2}},{\sqrt {3}}\right)&=\mathbb {Q} \left({\sqrt {2}}\right)\left({\sqrt {3}}\right)\\&=\left\{a+b{\sqrt {3}}\mid a,b\in \mathbb {Q} \left({\sqrt {2}}\right)\right\}\\&=\left\{a+b{\sqrt {2}}+c{\sqrt {3}}+d{\sqrt {6}}\mid a,b,c,d\in \mathbb {Q} \right\},\end{aligned}}

is an extension field of both $\mathbb {Q} ({\sqrt {2}})$ and $\mathbb {Q} ,$ of degree 2 and 4 respectively. It is also a simple extension, as one can show that

{\begin{aligned}\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})&=\mathbb {Q} ({\sqrt {2}}+{\sqrt {3}})\\&=\left\{a+b({\sqrt {2}}+{\sqrt {3}})+c({\sqrt {2}}+{\sqrt {3}})^{2}+d({\sqrt {2}}+{\sqrt {3}})^{3}\mid a,b,c,d\in \mathbb {Q} \right\}.\end{aligned}}

Finite extensions of $\mathbb {Q}$ are also calledalgebraic number fields and are important innumber theory. Another extension field of the rationals, which is also important in number theory, although not a finite extension, is the field ofp-adic numbers $\mathbb {Q} _{p}$ for a prime numberp.

It is common to construct an extension field of a given fieldK as aquotient ring of thepolynomial ringK[X] in order to "create" aroot for a given polynomialf(X). Suppose for instance thatK does not contain any elementx withx² = −1. Then the polynomial $X^{2}+1$ isirreducible inK[X], consequently theideal generated by this polynomial ismaximal, and $L=K[X]/(X^{2}+1)$ is an extension field ofK whichdoes contain an element whose square is −1 (namely theresidue class ofX).

By iterating the above construction, one can construct asplitting field of any polynomial fromK[X]. This is an extension fieldL ofK in which the given polynomial splits into a product of linear factors.

Ifp is anyprime number andn is a positive integer, there is a unique (up to isomorphism)finite field $GF(p^{n})=\mathbb {F} _{p^{n}}$ withpⁿ elements; this is an extension field of theprime field $\operatorname {GF} (p)=\mathbb {F} _{p}=\mathbb {Z} /p\mathbb {Z}$ withp elements.

Given a fieldK, we can consider the fieldK(X) of allrational functions in the variableX with coefficients inK; the elements ofK(X) are fractions of twopolynomials overK, and indeedK(X) is thefield of fractions of the polynomial ringK[X]. This field of rational functions is an extension field ofK. This extension is infinite.

Given aRiemann surfaceM, the set of allmeromorphic functions defined onM is a field, denoted by $\mathbb {C} (M).$ It is a transcendental extension field of $\mathbb {C}$ if we identify every complex number with the correspondingconstant function defined onM. More generally, given analgebraic varietyV over some fieldK, thefunction fieldK(V), consisting of the rational functions defined onV, is an extension field ofK.

Algebraic extension

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Main articles:Algebraic extension andAlgebraic element

An elementx of a field extension $L/K$ is algebraic overK if it is aroot of a nonzeropolynomial with coefficients inK. For example, ${\sqrt {2}}$ is algebraic over the rational numbers, because it is a root of $x^{2}-2.$ If an elementx ofL is algebraic overK, themonic polynomial of lowest degree that hasx as a root is called theminimal polynomial ofx. This minimal polynomial isirreducible overK.

An elements ofL is algebraic overK if and only if the simple extensionK(s) /K is a finite extension. In this case the degree of the extension equals the degree of the minimal polynomial, and a basis of theK-vector spaceK(s) consists of $1,s,s^{2},\ldots ,s^{d-1},$ whered is the degree of the minimal polynomial.

The set of the elements ofL that are algebraic overK form a subextension, which is called thealgebraic closure ofK inL. This results from the preceding characterization: ifs andt are algebraic, the extensionsK(s) /K andK(s)(t) /K(s) are finite. ThusK(s,t) /K is also finite, as well as the sub extensionsK(s ±t) /K,K(st) /K andK(1/s) /K (ifs ≠ 0). It follows thats ±t,st and 1/s are all algebraic.

Analgebraic extension $L/K$ is an extension such that every element ofL is algebraic overK. Equivalently, an algebraic extension is an extension that is generated by algebraic elements. For example, $\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})$ is an algebraic extension of $\mathbb {Q}$ , because ${\sqrt {2}}$ and ${\sqrt {3}}$ are algebraic over $\mathbb {Q} .$

A simple extension is algebraicif and only if it is finite. This implies that an extension is algebraic if and only if it is the union of its finite subextensions, and that every finite extension is algebraic.

Every fieldK has an algebraic closure, which isup to an isomorphism the largest extension field ofK which is algebraic overK, and also the smallest extension field such that every polynomial with coefficients inK has a root in it. For example, $\mathbb {C}$ is an algebraic closure of $\mathbb {R}$ , but not an algebraic closure of $\mathbb {Q}$ , as it is not algebraic over $\mathbb {Q}$ (for exampleπ is not algebraic over $\mathbb {Q}$ ).

Transcendental extension

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Main article:Transcendental extension

Given a field extension $L/K$ , a subsetS ofL is calledalgebraically independent overK if no non-trivial polynomial relation with coefficients inK exists among the elements ofS. The largest cardinality of an algebraically independent set is called thetranscendence degree ofL/K. It is always possible to find a setS, algebraically independent overK, such thatL/K(S) is algebraic. Such a setS is called atranscendence basis ofL/K. All transcendence bases have the same cardinality, equal to the transcendence degree of the extension. An extension $L/K$ is said to bepurely transcendental if and only if there exists a transcendence basisS of $L/K$ such thatL =K(S). Such an extension has the property that all elements ofL except those ofK are transcendental overK, but, however, there are extensions with this property which are not purely transcendental—a class of such extensions take the formL/K where bothL andK are algebraically closed.

IfL/K is purely transcendental andS is a transcendence basis of the extension, it doesn't necessarily follow thatL =K(S). On the opposite, even when one knows a transcendence basis, it may be difficult to decide whether the extension is purely separable, and if it is so, it may be difficult to find a transcendence basisS such thatL =K(S).

For example, consider the extension $\mathbb {Q} (x,y)/\mathbb {Q} ,$ where $x {\displaystyle x}$ is transcendental over $\mathbb {Q} ,$ and $y {\displaystyle y}$ is aroot of the equation $y^{2}-x^{3}=0.$ Such an extension can be defined as $\mathbb {Q} (X)[Y]/\langle Y^{2}-X^{3}\rangle ,$ in which $x {\displaystyle x}$ and $y {\displaystyle y}$ are theequivalence classes of $X {\displaystyle X}$ and $Y . {\displaystyle Y.}$ Obviously, the singleton set $\{x\}$ is transcendental over $\mathbb {Q}$ and the extension $\mathbb {Q} (x,y)/\mathbb {Q} (x)$ is algebraic; hence $\{x\}$ is a transcendence basis that does not generate the extension $\mathbb {Q} (x,y)/\mathbb {Q} (x)$ . Similarly, $\{y\}$ is a transcendence basis that does not generates the whole extension. However the extension is purely transcendental since, if one set $t=y/x,$ one has $x=t^{2}$ and $y=t^{3},$ and thus $t {\displaystyle t}$ generates the whole extension.

Purely transcendental extensions of an algebraically closed field occur asfunction fields ofrational varieties. The problem of finding arational parametrization of a rational variety is equivalent with the problem of finding a transcendence basis that generates the whole extension.

Normal, separable and Galois extensions

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An algebraic extension $L/K$ is callednormal if everyirreducible polynomial inK[X] that has a root inL completely factors into linear factors overL. Every algebraic extensionF/K admits a normal closureL, which is an extension field ofF such that $L/K$ is normal and which is minimal with this property.

An algebraic extension $L/K$ is calledseparable if the minimal polynomial of every element ofL overK isseparable, i.e., has no repeated roots in an algebraic closure overK. AGalois extension is a field extension that is both normal and separable.

A consequence of theprimitive element theorem states that every finite separable extension has a primitive element (i.e. is simple).

Given any field extension $L/K$ , we can consider itsautomorphism group ${\text{Aut}}(L/K)$ , consisting of all fieldautomorphismsα:L →L withα(x) =x for allx inK. When the extension is Galois this automorphism group is called theGalois group of the extension. Extensions whose Galois group isabelian are calledabelian extensions.

For a given field extension $L/K$ , one is often interested in the intermediate fieldsF (subfields ofL that containK). The significance of Galois extensions and Galois groups is that they allow a complete description of the intermediate fields: there is abijection between the intermediate fields and thesubgroups of the Galois group, described by thefundamental theorem of Galois theory.

Generalizations

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Field extensions can be generalized toring extensions which consist of aring and one of itssubrings. A closer non-commutative analog arecentral simple algebras (CSAs) – ring extensions over a field, which aresimple algebra (no non-trivial 2-sided ideals, just as for a field) and where thecenter 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 areBrauer equivalent to the reals or the quaternions. CSAs can be further generalized toAzumaya algebras, where the base field is replaced by a commutativelocal ring.

Extension of scalars

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Main article:Extension of scalars

Given a field extension, one can "extend scalars" on associated algebraic objects. For example, given a real vector space, one can produce a complex vector space viacomplexification. In addition to vector spaces, one can perform extension of scalars forassociative algebras defined over the field, such as polynomials orgroup algebras and the associatedgroup representations. 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 inextension of scalars: applications.

Notes

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^Fraleigh (1976, p. 293)
^Herstein (1964, p. 167)
^McCoy (1968, p. 116)
^Fraleigh (1976, p. 298)
^Herstein (1964, p. 193)
^Fraleigh (1976, p. 363)
^Fraleigh (1976, p. 319)
^Herstein (1964, p. 169)

References

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Fraleigh, John B. (1976),A First Course In Abstract Algebra (2nd ed.), Reading:Addison-Wesley,ISBN 0-201-01984-1
Herstein, I. N. (1964),Topics In Algebra, Waltham:Blaisdell Publishing Company,ISBN 978-1114541016{{citation}}:ISBN / Date incompatibility (help)
Lang, Serge (2004),Algebra,Graduate Texts in Mathematics, vol. 211 (Corrected fourth printing, revised third ed.), New York: Springer-Verlag,ISBN 978-0-387-95385-4
McCoy, Neal H. (1968),Introduction To Modern Algebra, Revised Edition, Boston:Allyn and Bacon,LCCN 68015225