Editing Field (mathematics) (section)

==== Residue fields ====
In addition to the field of fractions, which embeds {{math|''R''}} [[injective map|injectively]] into a field, a field can be obtained from a commutative ring {{math|''R''}} by means of a [[surjective map]] onto a field {{math|''F''}}. Any field obtained in this way is a [[quotient ring|quotient]] {{math|{{nowrap|''R'' / ''m''}}}}, where {{math|''m''}} is a [[maximal ideal]] of {{math|''R''}}. If {{math|''R''}} [[local ring|has only one maximal ideal]] {{math|''m''}}, this field is called the [[residue field]] of {{math|''R''}}.<ref>{{harvp|Eisenbud|1995|loc=p. 60}}</ref>

The [[principal ideal|ideal generated by a single polynomial]] {{math|''f''}} in the polynomial ring {{math|1=''R'' = ''E''[''X'']}} (over a field {{math|''E''}}) is maximal if and only if {{math|''f''}} is [[irreducible polynomial|irreducible]] in {{math|''E''}}, i.e., if {{math|''f''}} cannot be expressed as the product of two polynomials in {{math|''E''[''X'']}} of smaller [[degree of a polynomial|degree]]. This yields a field
: {{math|1=''F'' = ''E''[''X''] / ({{itco|''f''}}(''X'')).}}
This field {{math|''F''}} contains an element {{math|''x''}} (namely the [[residue class]] of {{math|''X''}}) which satisfies the equation
: {{math|1={{itco|''f''}}(''x'') = 0}}.
For example, {{math|'''C'''}} is obtained from {{math|'''R'''}} by [[adjunction (field theory)|adjoining]] the [[imaginary unit]] symbol {{mvar|i}}, which satisfies {{math|1={{itco|''f''}}(''i'') = 0}}, where {{math|1={{itco|''f''}}(''X'') = ''X''<sup>2</sup> + 1}}. Moreover, {{math|''f''}} is irreducible over {{math|'''R'''}}, which implies that the map that sends a polynomial {{math|{{itco|''f''}}(''X'') ∊ '''R'''[''X'']}} to {{math|{{itco|''f''}}(''i'')}} yields an isomorphism
: <math>\mathbf R[X]/\left(X^2 + 1\right) \ \stackrel \cong \longrightarrow \ \mathbf C.</math>