Field extension
In mathematics, particularly in algebra, a field extension is a pair of fields {\displaystyle E\subseteq F,}{\displaystyle E\subseteq F,} such that the operations of E are those of F restricted to E. In this case, F is an extension field of E and E is a subfield of F.[1][2][3] For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.
Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry.
1 Subfield
A subfield {\displaystyle K}K of a field {\displaystyle L}L is a subset {\displaystyle K\subseteq L}{\displaystyle K\subseteq L} that is a field with respect to the field operations inherited from {\displaystyle L}L. Equivalently, a subfield is a subset that contains {\displaystyle 1}1, and is closed under the operations of addition, subtraction, multiplication, and taking the inverse of a nonzero element of {\displaystyle K}K.
As 1 – 1 = 0, the latter definition implies {\displaystyle K}K and {\displaystyle L}L have the same zero element.
For example, the field of rational numbers is a subfield of the real numbers, which is itself a subfield of the complex numbers. More generally, the field of rational numbers is (or is isomorphic to) a subfield of any field of characteristic {\displaystyle 0}{\displaystyle 0}.
The characteristic of a subfield is the same as the characteristic of the larger field.
2 Extension field
If K is a subfield of L, then L is an extension field or simply extension of K, and this pair of fields is a field extension. Such a field extension is denoted L / K (read as “L over K”).
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 L / K.
Given a field extension L / K, the larger field L is a K-vector space. The dimension of this vector space is called the degree 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 a trivial extension. Extensions of degree 2 and 3 are called quadratic extensions and cubic extensions, respectively. A finite 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
{\displaystyle [M:K]=[M:L]\cdot [L:K].}{\displaystyle [M:K]=[M:L]\cdot [L:K].}
Given a field extension L / K and a subset S of L, there is a smallest subfield of L that contains K and S. It is the intersection of all subfields of L that contain K and S, and is denoted by K(S). One says that K(S) is the field generated by S over K, and that S is a generating set of K(S) over K. When {\displaystyle S={x_{1},\ldots ,x_{n}}}{\displaystyle S={x_{1},\ldots ,x_{n}}} is finite, one writes {\displaystyle K(x_{1},\ldots ,x_{n})}{\displaystyle K(x_{1},\ldots ,x_{n})} instead of {\displaystyle K({x_{1},\ldots ,x_{n}}),}{\displaystyle K({x_{1},\ldots ,x_{n}}),} and one says that K(S) is finitely generated over K. If S consists of a single element s, the extension K(s) / K is called a simple extension[4][5] and s is called a primitive element of the extension.[6]
An extension field of the form K(S) is often said to result from the adjunction of S to K.[7][8]
In characteristic 0, every finite extension is a simple extension. This is the primitive 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 of rational fractions in s over K.
3 Caveats
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.
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 ring homomorphism between two fields. Every non-zero ring homomorphism between fields is injective because fields do not possess nontrivial proper ideals, so field extensions are precisely the morphisms in the category of fields.
Henceforth, we will suppress the injective homomorphism and assume that we are dealing with actual subfields.
4 Examples
The field of complex numbers {\displaystyle \mathbb {C} }\mathbb{C} is an extension field of the field of real numbers {\displaystyle \mathbb {R} }\mathbb {R} , and {\displaystyle \mathbb {R} }\mathbb {R} in turn is an extension field of the field of rational numbers {\displaystyle \mathbb {Q} }\mathbb {Q} . Clearly then, {\displaystyle \mathbb {C} /\mathbb {Q} }{\displaystyle \mathbb {C} /\mathbb {Q} } is also a field extension. We have {\displaystyle [\mathbb {C} :\mathbb {R} ]=2}{\displaystyle [\mathbb {C} :\mathbb {R} ]=2} because {\displaystyle {1,i}}{1,i} is a basis, so the extension {\displaystyle \mathbb {C} /\mathbb {R} }{\displaystyle \mathbb {C} /\mathbb {R} } is finite. This is a simple extension because {\displaystyle \mathbb {C} =\mathbb {R} (i).}{\displaystyle \mathbb {C} =\mathbb {R} (i).} {\displaystyle [\mathbb {R} :\mathbb {Q} ]={\mathfrak {c}}}{\displaystyle [\mathbb {R} :\mathbb {Q} ]={\mathfrak {c}}} (the cardinality of the continuum), so this extension is infinite.
The field
{\displaystyle \mathbb {Q} ({\sqrt {2}})=\left{a+b{\sqrt {2}}\mid a,b\in \mathbb {Q} \right},}{\displaystyle \mathbb {Q} ({\sqrt {2}})=\left{a+b{\sqrt {2}}\mid a,b\in \mathbb {Q} \right},}
is an extension field of {\displaystyle \mathbb {Q} ,}{\displaystyle \mathbb {Q} ,} also clearly a simple extension. The degree is 2 because {\displaystyle \left{1,{\sqrt {2}}\right}}{\displaystyle \left{1,{\sqrt {2}}\right}} can serve as a basis.
The field
{\displaystyle {\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}}}{\displaystyle {\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 {\displaystyle \mathbb {Q} ({\sqrt {2}})}\mathbb{Q} ({\sqrt {2}}) and {\displaystyle \mathbb {Q} ,}{\displaystyle \mathbb {Q} ,} of degree 2 and 4 respectively. It is also a simple extension, as one can show that
{\displaystyle {\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}}}{\displaystyle {\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 {\displaystyle \mathbb {Q} }\mathbb {Q} are also called algebraic number fields and are important in number theory. Another extension field of the rationals, which is also important in number theory, although not a finite extension, is the field of p-adic numbers {\displaystyle \mathbb {Q} _{p}}{\displaystyle \mathbb {Q} _{p}} for a prime number p.
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 for a given polynomial f(X). Suppose for instance that K does not contain any element x with x2 = −1. Then the polynomial {\displaystyle X{2}+1}X2+1 is irreducible in K[X], consequently the ideal generated by this polynomial is maximal, and {\displaystyle L=K[X]/(X^{2}+1)}{\displaystyle L=K[X]/(X^{2}+1)} is an extension field of K which does contain an element whose square is −1 (namely the residue class of X).
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.
If p is any prime number and n is a positive integer, we have a finite field GF(pn) with pn elements; this is an extension field of the finite field {\displaystyle \operatorname {GF} §=\mathbb {Z} /p\mathbb {Z} }{\displaystyle \operatorname {GF} §=\mathbb {Z} /p\mathbb {Z} } with p elements.
Given a field K, we can consider the field K(X) of all rational functions in the variable X with coefficients in K; the elements of K(X) are fractions of two polynomials 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.
Given a Riemann surface M, the set of all meromorphic functions defined on M is a field, denoted by {\displaystyle \mathbb {C} (M).}{\displaystyle \mathbb {C} (M).} It is a transcendental extension field of {\displaystyle \mathbb {C} }\mathbb{C} if we identify every complex number with the corresponding constant function defined on M. More generally, given an algebraic variety V over some field K, then the function field of V, consisting of the rational functions defined on V and denoted by K(V), is an extension field of K.