Editing Surreal number (section)

== Surcomplex numbers ==

A '''surcomplex number''' is a number of the form {{math|''a'' + ''b''i}}, where {{mvar|a}} and {{mvar|b}} are surreal numbers and {{mvar|i}} is the square root of {{math|−1}}.<ref>[http://jamespropp.org/surreal/text.ps.gz Surreal vectors and the game of Cutblock], James Propp, August 22, 1994.</ref><ref name="Alling">{{cite book | last = Alling | first = Norman L. | title = Foundations of Analysis over Surreal Number Fields | publisher = North-Holland | series = Mathematics Studies 141 | year = 1987 | isbn = 0-444-70226-1}}</ref>  The surcomplex numbers form an [[algebraically closed field]] (except for being a proper class), [[isomorphic (mathematics)|isomorphic]] to the [[algebraic closure]] of the field generated by extending the [[rational numbers]] by a [[proper class]] of [[algebraically independent]] [[transcendental (mathematics)|transcendental]] elements. Up to field [[isomorphism]], this fact characterizes the field of surcomplex numbers within any fixed set theory.<ref name=Con01/>{{rp|Th.27}}