Axiomatic definition of complex numbers

Question

Trying to build axiomatically the set $\mathbb C$ of complex numbers, my first attempt was to define $\mathbb C$ with three structures: addition, multiplication and conjugate: $\langle\mathbb C,+,\times,{}^*\rangle$, with $\langle\mathbb C,+,\times\rangle$ forming a commutative field and the conjugate defined as a unitary operator such that (for all $a$, $b$ in the set) $(a+b)^*=a^*+b^*$, $(ab)^*=a^*b^*$, and ${a^*}^*=a$.

Corollary $0^*=0$ and $1^*=1$ for $0$ and $1$ respectively the modules of addition and multiplication.

Of course, the minimum set over which this structure can be build is $\mathbb Z_2$ with the usual addition and multiplication, and the conjugate equivalent to the identity.

If we require the conjugate not to be trivial (not to be the identity), then, by definition, we require 4 elements (the modules of addition and multiplication whose conjugates must be themselves), another element and its conjugate. I have found the following structure over $\{0,1,A,B\}$. $$ \begin{matrix} + & 0 & 1 & A & B \\ \hline 0 & 0 & 1 & A & B \\ 1 & 1 & 0 & B & A \\ A & A & B & 0 & 1 \\ B & B & A & 1 & 0 \\ \end{matrix} \qquad \begin{matrix} \times & 1 & A & B \\ \hline 1 & 1 & A & B \\ A & A & B & 1 \\ B & B & 1 & A \\ \end{matrix} \qquad \begin{matrix} 0^*=0 \\ 1^*=1 \\ A^*=B \\ B^*=A \\ \end{matrix} $$

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A special case of sets with the structure $\langle C,+,\times,{}^*\rangle$ are sets derived by the Cartesian product of a field $F$ with itself, by defining the addition by ordinate and the product $(a,b)\times(c,d):=(ac-bd,ad+bc)$, and the conjugate of $(a,b)$ as $(a,-b$). For this structure to be a field it is necessary that for any $x,y\in F$, then $xx+yy\neq 0$ (unles $x=y=0$). This means that ${\mathbb Z_3}^2$ can form a complex field, while ${\mathbb Z_5}^2$ cannot (at least not by this definition).

One way to ensure $xx+yy\ne0$ is having a total order in $F$ that is compatible with its addition and multiplication: that is that there is a positive subset $F_+\subset F$ in which addition and multiplication is closed, with $0\notin F_+$, and for any $x\ne0$ then $\{x,-x\}\cap F_+\ne\emptyset$. Given this structure it is easy to show that $xx$ is positive (and therefor non-zero) and $xx+yy$ is positive (and therefor non-zero).

The minimum field with positives are the rational numbers $\mathbb Q$, which means that we can build a complex field from $\mathbb Q^2$.

Note: in a field $F$ with positives $F_+$ the additive module $1$ must be positive and we can find an inductive subset $N$ (by having $n+1\in N$ if $n\in N$) which is analogous to the natural numbers $\mathbb N$. Completing the subfield derived by $N$ we must have a structure analogous to $\mathbb Q$ contained in $F$.

If we superimpose a convergence structure in $\mathbb Q^2$ then we will get to $\mathbb C$.

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However, I want to define $\mathbb C$ axiomatically and not by construction.

Question

Besides a field structure and a non-trivial conjugate, which other structures most be imposed to a set $C, \langle C,+,\times,{}^*\rangle$ so that we end up with $\mathbb C$?

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(I can define the subset $R$ of fixed points of $C$ by ${}^*$, and require positiveness in $R$ but somehow it seems like cheating.)

Answer 1

There is no first-order theory that will do the job. For if $T$ is a first-order theory over the language you describe, or an extension of it, and $T$ has $\mathbb{C}$ as a model, then by the Upward Lowenheim-Skolem Theorem, $T$ will have models of arbitrarily large cardinality. Moreover, if the language (number of non-logical symbols) is finite or countably infinite, then $T$ will have a model that is finite or countably infinite.

The "nearest" one can get along the lines you describe is the theory of algebraically closed fields of characteristic $0$. (We need an infinite number of axioms, but that's no problem.)