Complex Numbers of Type Rational a plus b root 2 form Field

Theorem

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

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

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

Then the algebraic structure:

$\struct {\Q \sqbrk {\sqrt 2}, +, \times}$

where $+$ and $\times$ are conventional addition and multiplication on real numbers, is a number field.

Proof

From Real Numbers of Type Rational a plus b root 2 form Field, the set:

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

forms a field.

As $\Q \sqbrk {\sqrt 2} \subseteq \R$ and $\R \subseteq \C$ it follows that $\struct {\Q \sqbrk {\sqrt 2}, +, \times}$ is a subfield of $\C$.

Hence the result by definition of number field.

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $3$: Field Theory: Definition and Examples of Field Structure: $\S 88 \beta$