Editing Field (mathematics) (section)

==== Transcendence bases ====
The above-mentioned field of [[rational fraction]]s {{math|''E''(''X'')}}, where {{math|''X''}} is an [[indeterminate (variable)|indeterminate]], is not an algebraic extension of {{math|''E''}} since there is no polynomial equation with coefficients in {{math|''E''}} whose zero is {{math|''X''}}. Elements, such as {{math|''X''}}, which are not algebraic are called [[Algebraic element|transcendental]]. Informally speaking, the indeterminate {{math|''X''}} and its powers do not interact with elements of {{math|''E''}}. A similar construction can be carried out with a set of indeterminates, instead of just one.

Once again, the field extension {{math|''E''(''x'') / ''E''}} discussed above is a key example: if {{math|''x''}} is not algebraic (i.e., {{math|''x''}} is not a [[root of a function|root]] of a polynomial with coefficients in {{math|''E''}}), then {{math|''E''(''x'')}} is isomorphic to {{math|''E''(''X'')}}. This isomorphism is obtained by substituting {{math|''x''}} to {{math|''X''}} in rational fractions.

A subset {{math|''S''}} of a field {{math|''F''}} is a [[transcendence basis]] if it is [[algebraically independent]] (do not satisfy any polynomial relations) over {{math|''E''}} and if {{math|''F''}} is an algebraic extension of {{math|''E''(''S'')}}. Any field extension {{math|''F'' / ''E''}} has a transcendence basis.<ref>{{harvp|Bourbaki|1988|loc=Chapter V, §14, No. 2, Theorem 1}}</ref> Thus, field extensions can be split into ones of the form {{math|''E''(''S'') / ''E''}} ([[transcendental extension|purely transcendental extensions]]) and algebraic extensions.