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It's well known that the surreal numbers $\mathbf{No}$ are the largest ordered "field" (more accurately, they form a proper class with field structure, which is sometimes called a Field with capital F), in the sense that every other ordered field can be embedded in them. Since the surreal numbers are real closed, their algebraic closure is given by $\mathbf{No}[i]$, the surcomplex numbers.

My question is, is there a similar characterization of the surcomplex numbers as the largest algebraically closed Field of characteristic zero? If they aren't the largest, is it possible to find a proper class with that property?

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    $\begingroup$ If $F$ is an algebraically closed field of characteristic zero, and $\kappa$ is the cardinal of a transcendance basis of $F$ over $\mathbb{Q}$, then $F$ is isomorphic to the relative algebraic closure of the smallest subfield of $No[i]$ containing $\mathbb{Q}$ and the $\omega^{\omega^{\alpha}}$ for $\alpha < \kappa$, so yes. $\endgroup$
    – nombre
    Commented Jan 22, 2017 at 23:49
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    $\begingroup$ @nombre Wow, thank you! I didn't expect the answer to be that simple. If you post it as an answer I'll accept it. $\endgroup$
    – pregunton
    Commented Jan 23, 2017 at 0:19

1 Answer 1

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Let $F$ be an algebraically closed field of characteristic zero.

Let $\kappa$ be the cardinal of a transcendance basis of $F$ over its prime subfield $\mathbb{Q}$.

Since the ${\omega_0}^{{\omega_0}^{\alpha}}$, $\alpha < \kappa$ satisfy the relation $\forall \alpha < \beta < \kappa, \forall n \in \mathbb{N}, n.({\omega_0}^{{\omega_0}^{\alpha}})^n < {\omega_0}^{{\omega_0}^{\beta}}$, they form an algebraically independant familly over $\mathbb{Q} \subset No$.

So $No[i]$, being algebraically closed, contains an isomorphic copy of $F$ as the relative algebraic closure of $\mathbb{Q}(({\omega_0}^{{\omega_0}^{\alpha}})_{\alpha < \kappa})$.


This is also true in NBG with global choice if $F$ is an algebraically closed Field of characteristic zero, and to see this one only needs to repeat the same argument with $Ord$ instead of $\kappa$.

Note that being a universal algebraically closed Field of characteristic zero caracterises $No[i]$ as a field whereas being a universal real closed Field does not characterise $No$. The theory of algebraically closed fields of a given characteristic is indeed more stable than that of real closed fields.

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