Quadratic Integers over 2 form Ordered Integral Domain

Theorem
Let $\Z \sqbrk {\sqrt 2}$ denote the set of quadratic integers over $2$:
 * $\Z \sqbrk {\sqrt 2} := \set {a + b \sqrt 2: a, b \in \Z}$

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

Then the algebraic structure:
 * $\struct {\Z \sqbrk {\sqrt 2}, +, \times}$

where $+$ and $\times$ are conventional addition and multiplication on real numbers, is an ordered integral domain.

Proof
We have that Quadratic Integers over 2 form Subdomain of Reals.

We also have that such numbers are real.

The result follows from Real Numbers form Ordered Integral Domain.