Cyclotomic Ring/Examples/5th

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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:

\(\displaystyle z_1 - z_2\) \(=\) \(\displaystyle \paren {a_1 + i b_1 \sqrt 5} - \paren {a_2 + i b_2 \sqrt 5}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {a_1 - a_2} + i \paren {b_1 - b_2} \sqrt 5\) Definition of Complex Addition
\(\displaystyle \) \(\in\) \(\displaystyle \Z \sqbrk {i \sqrt 5}\) as $a_1 - a_2$ and $b_1 - b_2$ are both integers


and:

\(\displaystyle z_1 z_2\) \(=\) \(\displaystyle \paren {a_1 + i b_1 \sqrt 5} \paren {a_2 + i b_2 \sqrt 5}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {a_1 a_2 - 5 b_1 b_2} + i \sqrt 5 \paren {a_1 b_2 + a_2 b_1}\) Definition of Complex Multiplication
\(\displaystyle \) \(\in\) \(\displaystyle \Z \sqbrk {i \sqrt 5}\) as $a_1 a_2 - 5 b_1 b_2$ and $a_1 b_2 + a_2 b_1$ are both integers

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

$\blacksquare$


Sources