Editing Field extension (section)

==Subfield==
A '''subfield''' <math>K</math> of a [[field (mathematics)|field]] <math>L</math> is a [[subset]] <math>K\subseteq L</math> that is a field with respect to the field operations inherited from <math>L</math>. Equivalently, a subfield is a subset that contains the [[multiplicative identity]] <math>1</math>, and is [[Closure (mathematics)|closed]] under the operations of addition, subtraction, multiplication, and taking the [[multiplicative inverse|inverse]] of a nonzero element of <math>K</math>.

As {{math|1=1 – 1 = 0}}, the latter definition implies <math>K</math> and <math>L</math> have the same [[zero element]].

For example, the field of [[rational number]]s is a subfield of the [[real number]]s, which is itself a subfield of the complex numbers. More generally, the field of rational numbers is (or is [[isomorphism|isomorphic]] to) a subfield of any field of [[characteristic of a ring|characteristic]] <math>0</math>.

The [[characteristic (algebra)|characteristic]] of a subfield is the same as the characteristic of the larger field.