Existence and Uniqueness of the Complex Numbers

Let $\mathbb{R}$ be the set of real numbers, with its addition, its multiplication and its order.

Call a pair $(K,j)$, consisting of a field $K$ and an element $j\in K$, a complex pair if the following three conditions hold, where $+$ and $\cdot$ denote the addition and multiplication of $K$ and $1$ denotes the multiplicative identity of $K$:

(a) $\mathbb{R}\subseteq K$, and for all $a,b\in\mathbb{R}$ the sum $a+b$ and the product $a\cdot b$ formed in $K$ coincide with the sum and product of $a$ and $b$ formed in $\mathbb{R}$;

(b) $j\cdot j=-1$, where $-1$ is the additive inverse of $1$ in $K$;

(c) for every $z\in K$ there exist $a,b\in\mathbb{R}$ with $z=a+b\cdot j$.

Then the following hold.

1. (Existence) There exists a complex pair.

2. (Uniqueness up to a unique isomorphism) Let $(K,j)$ and $(K',j')$ be complex pairs. Then there is exactly one map $\varphi:K\to K'$ such that

$\varphi(a)=a$ for every $a\in\mathbb{R}$, and $\varphi(j)=j'$. This map is a bijection, and its inverse has the corresponding properties with the roles of $(K,j)$ and $(K',j')$ exchanged.