Quadratic Integers over 2 form Ordered Integral Domain

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, is an ordered integral domain.

Proof
We have that Numbers of Type Integer a plus b root 2 form Subdomain of Reals.

We also have that such numbers are real.

The result follows from Real Numbers form Ordered Integral Domain.