Editing Vector space (section)

===Complex numbers and other field extensions===
The set of [[complex numbers]] {{math|'''C'''}}, numbers that can be written in the form {{math|1=''x'' + ''iy''}} for [[real numbers]] {{math|''x''}} and {{math|''y''}} where {{math|''i''}} is the [[imaginary unit]], form a vector space over the reals with the usual addition and multiplication: {{math|1=(''x'' + ''iy'') + (''a'' + ''ib'') = (''x'' + ''a'') + ''i''(''y'' + ''b'')}} and {{math|1=''c'' ⋅ (''x'' + ''iy'') = (''c'' ⋅ ''x'') + ''i''(''c'' ⋅ ''y'')}} for real numbers {{math|''x''}}, {{math|''y''}}, {{math|''a''}}, {{math|''b''}} and {{math|''c''}}.  The various axioms of a vector space follow from the fact that the same rules hold for complex number arithmetic. The example of complex numbers is essentially the same as (that is, it is ''isomorphic'' to) the vector space of ordered pairs of real numbers mentioned above: if we think of the complex number {{math|''x'' + ''i'' ''y''}} as representing the ordered pair {{math|(''x'', ''y'')}} in the [[complex plane]] then we see that the rules for addition and scalar multiplication correspond exactly to those in the earlier example.

More generally, [[field extension]]s provide another class of examples of vector spaces, particularly in algebra and [[algebraic number theory]]: a field {{math|''F''}} containing a [[Field extension|smaller field]] {{math|''E''}} is an {{math|''E''}}-vector space, by the given multiplication and addition operations of {{math|''F''}}.{{sfn|Lang|2002|loc = ch. V.1}} For example, the complex numbers are a vector space over {{math|'''R'''}}, and the field extension <math>\mathbf{Q}(i\sqrt{5})</math> is a vector space over {{math|'''Q'''}}.  <!--A particularly interesting type of field extension in [[number theory]] is {{math|'''Q'''(''α'')}}, the extension of the rational numbers {{math|'''Q'''}} by a fixed complex number {{math|''α''}}. {{math|'''Q'''(''α'')}} is the smallest field containing the rationals and a fixed complex number ''α''. Its dimension as a vector space over {{math|'''Q'''}} depends on the choice of {{math|''α''}}.-->