Editing Field (mathematics) (section)

==== Algebraic extensions ====
A pivotal notion in the study of field extensions {{math|''F'' / ''E''}} are [[algebraic element]]s. An element {{math|''x'' ∈ ''F''}} is ''algebraic'' over {{mvar|E}} if it is a [[zero of a function|root]] of a [[polynomial]] with [[coefficient]]s in {{mvar|E}}, that is, if it satisfies a [[polynomial equation]]
: {{math|1=''e''<sub>''n''</sub>&hairsp;''x''<sup>''n''</sup> + ''e''<sub>''n''−1</sub>''x''<sup>''n''−1</sup> + ⋯ + ''e''<sub>1</sub>''x'' + ''e''<sub>0</sub> = 0}},
with {{math|''e''<sub>''n''</sub>, ..., ''e''<sub>0</sub>}} in {{mvar|E}}, and {{math|''e''<sub>''n''</sub> ≠ 0}}.
For example, the [[imaginary unit]] {{math|''i''}} in {{math|'''C'''}} is algebraic over {{math|'''R'''}}, and even over {{math|'''Q'''}}, since it satisfies the equation
: {{math|1=''i''<sup>2</sup> + 1 = 0}}.
A field extension in which every element of {{math|''F''}} is algebraic over {{math|''E''}} is called an [[algebraic extension]]. Any finite extension is necessarily algebraic, as can be deduced from the above multiplicativity formula.<ref>{{harvp|Artin|1991|loc=Corollary 13.3.6}}</ref>

The subfield {{math|''E''(''x'')}} generated by an element {{math|''x''}}, as above, is an algebraic extension of {{math|''E''}} if and only if {{math|''x''}} is an algebraic element. That is to say, if {{math|''x''}} is algebraic, all other elements of {{math|''E''(''x'')}} are necessarily algebraic as well. Moreover, the degree of the extension {{math|''E''(''x'') / ''E''}}, i.e., the dimension of {{math|''E''(''x'')}} as an {{math|''E''}}-vector space, equals the minimal degree {{math|''n''}} such that there is a polynomial equation involving {{math|''x''}}, as above. If this degree is {{math|''n''}}, then the elements of {{math|''E''(''x'')}} have the form
: <math>\sum_{k=0}^{n-1} a_k x^k, \ \ a_k \in E.</math>

For example, the field {{math|'''Q'''(''i'')}} of [[Gaussian rational]]s is the subfield of {{math|'''C'''}} consisting of all numbers of the form {{math|''a'' + ''bi''}} where both {{math|''a''}} and {{math|''b''}} are rational numbers: summands of the form {{math|''i''<sup>2</sup>}} (and similarly for higher exponents) do not have to be considered here, since {{math|''a'' + ''bi'' + ''ci''<sup>2</sup>}} can be simplified to {{math|''a'' − ''c'' + ''bi''}}.