Ring of Polynomials over Reals is not Field

Theorem
Let $\R \sqbrk X$ be the ring of polynomials in an indeterminate $X$ over $\R$.

Then $\R \sqbrk X$ is not a field.

Proof
Consider the polynomial $x + 1$ in $\R \sqbrk X$.

There exists no polynomial $\map f x$ such that:
 * $\paren {x + 1} \map f x = 1$

This is because the has degree $1$, and the  has degree $0$.