Real Numbers of Type Rational a plus b root 2 form Field/Corollary

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

... that is, all numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are rational numbers. The field $\left({\Q \left[{\sqrt 2}\right], +, \times}\right)$ is a subfield of $\left({\R, +, \times}\right)$.

Proof
So $\Q \left[{\sqrt 2}\right] \subseteq \R$.

From Numbers of Type Rational a plus b root 2 form Field, $\left({\Q \left[{\sqrt 2}\right], +, \times}\right)$ is a field.

As stated in the proof of the Numbers of Type Rational a plus b root 2 form Field, numbers of the form $a + b \sqrt 2$ are real.

Hence the result by definition of subfield.