A surcomplex number is a number of the form , where a and b are surreal numbers. The surcomplex numbers form an algebraically closed field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

, isomorphic to the algebraic closure

Algebraic closure

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....

 of the field generated by extending the rational numbers by a proper class of algebraically independent transcendental elements. Up to field isomorphism

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

, this fact characterizes the field of surcomplex numbers within any fixed set theory.