Editing Field extension (section)

== Examples ==
The field of complex numbers <math>\Complex</math> is an extension field of the field of [[real number]]s <math>\R</math>, and <math>\R</math> in turn is an extension field of the field of rational numbers <math>\Q</math>. Clearly then, <math>\Complex/\Q</math> is also a field extension. We have <math>[\Complex:\R] =2</math> because <math>\{1, i\}</math> is a basis, so the extension <math>\Complex/\R</math> is finite. This is a simple extension because <math>\Complex = \R(i).</math> <math>[\R:\Q] =\mathfrak c</math> (the [[cardinality of the continuum]]), so this extension is infinite.

The field 

:<math>\Q(\sqrt{2}) =  \left \{ a + b\sqrt{2} \mid a,b \in \Q \right \},</math>

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

The field

:<math>\begin{align}
\Q\left(\sqrt{2}, \sqrt{3}\right) &= \Q \left(\sqrt{2}\right) \left(\sqrt{3}\right) \\
&= \left\{ a+b\sqrt{3} \mid a,b \in \Q\left(\sqrt{2}\right) \right\} \\
&= \left\{ a + b \sqrt{2} + c\sqrt{3} + d\sqrt{6} \mid a,b,c, d \in \Q \right\},
\end{align}</math>

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

:<math>\begin{align}
\Q(\sqrt{2}, \sqrt{3}) &= \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 \Q\right\}.
\end{align}</math>

Finite extensions of <math>\Q</math> are also called [[algebraic number field]]s 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 number]]s <math>\Q_p</math> 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 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 <math>X^2+1</math> is [[irreducible polynomial|irreducible]] in ''K''[''X''], consequently the [[ideal (ring theory)|ideal]] generated by this polynomial is [[maximal ideal|maximal]], and <math>L = K[X]/(X^2+1)</math> is an extension field of ''K'' which ''does'' contain an element whose square is −1 (namely the [[modular arithmetic|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, there is a unique (up to isomorphism) [[finite field]] <math>GF(p^n) = \mathbb{F}_{p^n}</math> with ''p<sup>n</sup>'' elements; this is an extension field of the [[prime field]] <math>\operatorname{GF}(p) = \mathbb{F}_p = \Z/p\Z</math> with ''p'' elements.

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.

Given a [[Riemann surface]] ''M'', the set of all [[meromorphic function]]s defined on ''M'' is a field, denoted by <math>\Complex(M).</math> It is a transcendental extension field of <math>\Complex</math> 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'', the [[function field of an algebraic variety|function field]] ''K''(''V''), consisting of the rational functions defined on ''V'', is an extension field of ''K''.