I was looking for a complex extension of the surreals and then I found $\operatorname{No}[i]$, what are its properties and how do I express $x+yi$ in the $a | b$ notation?
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2 Answers
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The surreal complex field $\text{No}[i]$, known as the surcomplex field, is a proper-class-sized set-saturated algebraically closed field. It is universal for all fields of characteristic 0. Indeed, under global choice, every class field structure in characteristic 0 embeds as a subfield of $\text{No}[i]$. It is the algebraic closure of the transcendental field extension of $\mathbb{Q}$ by a proper class of algebraically independent transcendentals.
answered Jul 20, 2023 at 14:09
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1$\begingroup$ Meanwhile, I don't know if there is a useful Conway game representation of the surreal complex numbers, and I shall be interested to read about this if there is such a representation. $\endgroup$ Commented Jul 20, 2023 at 14:35
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1$\begingroup$ Btw I've been experimenting with the "surquaternions" which are a quaternionic extension to the surreals (can be written as $\operatorname{No}[i, j]$, or using Alec Rhea 's notation $\operatorname{N_o}[i, j]$ $\endgroup$ Commented Sep 1, 2023 at 21:52
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1$\begingroup$ Why is it not sufficient to play Conway’s game in the real and imaginary parts separately to represent the whole field? / is that just considered uninteresting? And something is sought which isn’t playing 2 games at once but a single game. $\endgroup$ Commented Sep 2, 2023 at 0:28
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1$\begingroup$ Of course we can do that, but what I was wondering is whether we can find a collection of Conway games, just as they are (not pairs of games or what have you), which admit a field structure making them the surreal complex numbers. $\endgroup$ Commented Sep 2, 2023 at 0:44
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$\begingroup$ @SidharthGhosal So that could be written using set theory $\left\{ \left\{ \Re(a+bi)_L | \Re(a+bi)_R \right\}, \left\{ \Im(a+bi)_L | \Im(a+bi)_R \right\} \right\}$ $\endgroup$ Commented Jan 2 at 20:51
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In addition to the properties mentioned in Joel's answer, $N_0[i]$ is a cogenerator in the category of all fields of characteristic $0$ by a similar argument to the one given by Keith Kearnes here, modulo the appropriate foundation to squeeze them into a category.
answered Jul 20, 2023 at 14:31
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2$\begingroup$ I'd always thought it was $\mathit{No}$ or $\mathrm{No}$, as in @JoelDavidHamkins's answer, for "nombre" or something like that, but viewing it as a particularly important characteristic-$0$ "field" does make sense of the notation $N_0$. Is that the usual notation? $\endgroup$– LSpiceCommented Jul 20, 2023 at 15:24
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3$\begingroup$ @LSpice I'm not sure where I first encountered this notation (maybe in 'Foundations of Surreal Analysis' by Alling?), but I believe it was a reversal of $O_n$ for the ordinals. $\endgroup$ Commented Jul 20, 2023 at 17:10
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3$\begingroup$ I don't think I've ever seen $O_n$ as a notation for the ordinals, although I have certainly seen On as well as Ord. I always took No as the notation for the surreals to consist of its inversion of On, as well as the fact that "No." is a common abbreviation for "Number". $\endgroup$ Commented Jul 20, 2023 at 19:28
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2$\begingroup$ @JoelDavidHamkins I trust your breadth of literature exposure; perhaps this is just a contrivance of mine, seeing the On and No duality and allowing my propensity for subscripts to take over. $\endgroup$ Commented Jul 20, 2023 at 20:39
Nospaces poorly. If you want something similar looking, you can use $\mathit{No}$\mathit{No}; I edited accordingly. If you would like it upright, as in @JoelDavidHamkins's answer, then you can use $\mathrm{No}$\mathrm{No}(but note that\operatornameis often the better tool for upright mathematics—it is meant, as the name suggests, for "operators", like $\sin$, $\cos$, $\lim$, etc). $\endgroup$