Cyclotomic Ring/Examples/5th

Examples of Cyclotomic Rings
The $5$th cyclotomic ring is the algebraic structure:
 * $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$

where $\Z \sqbrk {i \sqrt 5}$ is the set $\set {a + i b \sqrt 5: a, b \in \Z}$.

$\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ is a ring.

Proof
We have that $\Z \sqbrk {i \sqrt 5}$ is a subset of the Field of Complex Numbers $\struct {\C, +, \times}$.

So to prove that $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ is a ring it is sufficient to demonstrate that $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ fulfils the conditions of the Subring Test.

First we note that setting $a = 1, b = 0$ we have that $1 + 0 i \in \Z \sqbrk {i \sqrt 5}$ and so $\Z \sqbrk {i \sqrt 5} \ne \O$.

Let $z_1 = a_1 + i b_1 \sqrt 5$ and $z_2 = a_2 + i b_2 \sqrt 5$ be arbitrary elements of $\Z \sqbrk {i \sqrt 5}$

Then:

and:

The Subring Test is satisfied, and so $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ is a ring.