Quadratic Integers over 2 form Subdomain of Reals/Proof 2

Proof
From Integers form Subdomain of Reals, $\struct {\Z, +, \times}$ is an integral subdomain of the real numbers $\R$.

We have that $\sqrt 2 \in \R$.

Every expression of the form:
 * $a_0 + a_1 \sqrt 2 + a_2 \paren {\sqrt 2}^2 + \cdots + a_n \paren {\sqrt 2}^n$

can be simplified to a number of the form $a + b \sqrt 2$, where $a, b \in \Z$.

The result follows from Set of Polynomials over Integral Domain is Subring.