Field of Quotients of Ring of Polynomial Forms on Reals that yields Complex Numbers

Theorem
Let $\struct {\R, +, \times}$ denote the field of real numbers.

Let $X$ be transcendental over $\R$.

Let $\R \sqbrk X$ be the ring of polynomials in $X$ over $F$.

Consider the quotient field:
 * $\R \sqbrk X / \ideal p$

where:
 * $p = X^2 + 1$
 * $\ideal p$ denotes the ideal generated by $p$.

Then $\R \sqbrk X / \ideal p$ is the field of complex numbers.

Proof
It is taken as read that $X^2 + 1$ is irreducible in $\R \sqbrk X$.

Hence by Polynomial Forms over Field form Principal Ideal Domain: Corollary 1, $\R \sqbrk X / \ideal p$ is indeed a field.

Let $\nu$ be the quotient epimorphism from $\R \sqbrk X$ onto $\R \sqbrk X / \ideal p$.

From Quotient Ring Epimorphism is Epimorphism:
 * $\map \ker {\nu \vert \R} = \R \cap \ideal p = \set 0$