Quadratic Integers over 3 form Integral Domain
Theorem
Let $\R$ denote the set of real numbers.
Let $\Z \sqbrk {\sqrt 3} \subseteq \R$ denote the set of quadratic integers over $3$:
- $\Z \sqbrk {\sqrt 3} = \set {a + b \sqrt 3: a, b \in \Z}$
Then $\struct {\Z \sqbrk {\sqrt 3}, +, \times}$ is an integral domain.
Proof
From Real Numbers form Integral Domain we have that $\struct {\R, +, \times}$ is an integral domain.
Hence to demonstrate that $\struct {\Z \sqbrk {\sqrt 3}, +, \times}$ is an integral domain, we can use the Subdomain Test.
We have that the unity of $\struct {\R, +, \times}$ is $1$.
Then we note:
- $1 = 1 + 0 \times \sqrt 3$
and so $1 \in S$.
Thus property $(2)$ of the Subdomain Test is fulfilled.
It remains to demonstrate that $\struct {\Z \sqbrk {\sqrt 3}, +, \times}$ is a subring of $\struct {\R, +, \times}$, so fulfilling property $(2)$ of the Subdomain Test.
Hence we use the Subring Test.
We note that $\Z \sqbrk {\sqrt 3} \ne \O$ as $1 \in \Z \sqbrk {\sqrt 3}$.
This fulfils property $(1)$ of the Subring Test.
Let $x, y \in \Z \sqbrk {\sqrt 3}$ such that:
- $x = a + b \sqrt 3$
- $y = c + d \sqrt 3$
Then:
\(\ds x + \paren {-y}\) | \(=\) | \(\ds \paren {a + b \sqrt 3} - \paren {c + d \sqrt 3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a - c} + \paren {b \sqrt 3 - d \sqrt 3}\) | Definition of Real Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a - c} + \paren {b - d} \sqrt 3\) | ||||||||||||
\(\ds \) | \(\in\) | \(\ds \Z \sqbrk {\sqrt 3}\) |
This fulfils property $(2)$ of the Subring Test.
Then:
\(\ds x \times y\) | \(=\) | \(\ds \paren {a + b \sqrt 3} \paren {c + d \sqrt 3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a c + a d \sqrt 3 + b c \sqrt 3 + 3 b d\) | Definition of Real Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a c + 3 b d} + \paren {a d + b c} \sqrt 3\) | ||||||||||||
\(\ds \) | \(\in\) | \(\ds \Z \sqbrk {\sqrt 3}\) |
This fulfils property $(3)$ of the Subring Test.
Hence the result.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 6$: Isomorphisms of Algebraic Structures: Exercise $6.7 \ \text {(a)}$
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous): Exercises: Chapter $1$: Exercise $1 \ \text{(iv)}$