Quadratic Integers over 2 form Subdomain of Reals/Proof 2

Theorem
Let $\Z \left[{\sqrt 2}\right]$ denote the set:
 * $\Z \left[{\sqrt 2}\right] := \left\{{a + b \sqrt 2: a, b \in \Z}\right\}$

... that is, all numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are integers.

Then the algebraic structure:
 * $\left({\Z \left[{\sqrt 2}\right], +, \times}\right)$

where $+$ and $\times$ are conventional addition and multiplication on real numbers, form an integral subdomain of the real numbers $\R$.

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

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

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

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.