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

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

Let $\Q \sqbrk {\sqrt 2}$ denote the set:

$\Q \sqbrk {\sqrt 2} := \set {a + b \sqrt 2: a, b \in \Q}$

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

The field $\struct {\Q \sqbrk {\sqrt 2}, +, \times}$ is a subfield of $\struct {\R, +, \times}$.

Proof

So $\Q \sqbrk {\sqrt 2} \subseteq \R$.

From Real Numbers of Type Rational a plus b root 2 form Field, $\struct {\Q \sqbrk {\sqrt 2}, +, \times}$ is a field.

As stated in the proof of the Real 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.

$\blacksquare$

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $4$: Fields: $\S 16$. Subfields: Example $22$