# Characterizing the surcomplex numbers

Conway showed that the Field of surreal numbers ("${\bf No}$") is the maximal totally ordered Field.

Later Jacob Lurie showed that the Group of all partizan games ${\bf Pg}$ is the universally embedding partially ordered Abelian Group.

Is there some analogous functorial characterization of the Field of surcomplex numbers ${\bf No}[i]$?

Or might there be some sense in which ${\bf No}[i]$ isn't the "right" algebraic closure of ${\bf No}$? (Recall what happens when one takes the algebraic closure of the field of $p$-adic numbers: one gets a system that is unsatisfactory because it is not metrically complete, and then one has to pass to an even larger system to obtain the correct $p$-adic analogue of the field of complex numbers. Of course this is a vague analogy; in particular, the notion of metric completeness is not relevant in the case of ${\bf No}[i]$.)

Come to think of it, why is ${\bf No}[i]$ algebraically closed?

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The property that adjoining $i$ makes your field algebraically closed is called being real closed and has many, many equivalent formulations: en.wikipedia.org/wiki/Real_closed_field –  Qiaochu Yuan May 3 '11 at 5:59
Thanks! I see that Norman Alling, in his book "Foundations of Analysis over Surreal Number Fields" (which I obtained after posting my query), proves that the surreal numbers are real-closed by identifying them with formal power series of a suitable kind. He does not (as far as I can tell) characterize ${\bf Cx}$ functorially. Perhaps the right functorial characterization would involve fields equipped with an involution (complex conjugation) that satisfies various properties and in particular induces partial orderings based on real part, imaginary part, and modulus. –  James Propp May 3 '11 at 16:44