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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?

SebbyIsSwag
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    TeX note: $No$ No spaces 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 \operatorname is often the better tool for upright mathematics—it is meant, as the name suggests, for "operators", like $\sin$, $\cos$, $\lim$, etc). – LSpice Jul 20 '23 at 15:22
  • Elements are not given in ${a|b}$ notation. – Gerald Edgar Jul 20 '23 at 16:50
  • @GeraldEdgar can't we write $a+bi$ as $a_L+b_Li | a_R+b_Ri$? – SebbyIsSwag Jul 20 '23 at 18:11
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    @SebbyIsSwag The Conway game notation realizes surreal numbers as Conway games (which simply drop the requirement that members of the left set must be less than members of the right set). And those games have a kind of algebra. To my way of thinking, the question would be: can we realize the surreal complex numbers in the space of Conway games? For example, which game is the imaginary unit $i$? Your proposed representation does provides a notation for the surreal complex numbers, but not as Conway games. – Joel David Hamkins Jul 20 '23 at 20:45
  • For general Conway games, multiplication is undefined. – Gerald Edgar Sep 02 '23 at 01:13

2 Answers2

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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.

Joel David Hamkins
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    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. – Joel David Hamkins Jul 20 '23 at 14:35
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    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 $\operatorname{N_o}[i, j]$ – SebbyIsSwag Sep 01 '23 at 21:52
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    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. – Sidharth Ghoshal Sep 02 '23 at 00:28
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    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. – Joel David Hamkins Sep 02 '23 at 00:44
  • @SidharthGhosal So that could be written using set theory $\left{ \left{ \Re(a+bi)_L | \Re(a+bi)_R \right}, \left{ \Im(a+bi)_L | \Im(a+bi)_R \right} \right}$ – SebbyIsSwag Jan 02 '24 at 20:51
  • @JoelDavidHamkins Maybe something like Complex Nim: 3 Heaps of objects: Heap A, Heap B and Heap C. Player take turns and in each you take one object off. The complex value of the position is given by A + Bi + C (where A, B and C are counts of the heaps). The player makes the heap reach 0 loses. – SebbyIsSwag Jan 02 '24 at 20:59
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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.

Alec Rhea
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    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? – LSpice Jul 20 '23 at 15:24
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    @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. – Alec Rhea Jul 20 '23 at 17:10
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    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". – Joel David Hamkins Jul 20 '23 at 19:28
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    @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. – Alec Rhea Jul 20 '23 at 20:39