Quadratic Integers over 2 form Subdomain of Reals/Proof 2

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, form an integral subdomain of the real numbers $\R$.

Proof

From Integers form Subdomain of Reals, $\struct {\Z, +, \times}$ is an integral subdomain of the real numbers $\R$.

We have that $\sqrt 2 \in \R$.

Every expression of the form:

$a_0 + a_1 \sqrt 2 + a_2 \paren {\sqrt 2}^2 + \cdots + a_n \paren {\sqrt 2}^n$

can be simplified to a number of the form $a + b \sqrt 2$, where $a, b \in \Z$.

The result follows from Set of Polynomials over Integral Domain is Subring.

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 64.1$ Polynomial rings over an integral domain: Illustration