Irreducible Elements of 5th Cyclotomic Ring

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Theorem

Let $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ denote the $5$th cyclotomic ring.


The following elements of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ are irreducible:

$2$
$3$
$1 + i \sqrt 5$
$1 - i \sqrt 5$


Proof


Let $z = x + i y$ be an element of $\Z \sqbrk {i \sqrt 5}$ in the set $S$, where:

$S := \set {2, 3, 1 + i \sqrt 5, 1 - i \sqrt 5}$

Let $z$ have a non-trivial factorization:

$z = z_1 z_2$

where neither $z_1$ nor $z_2$ are units of $\Z \sqbrk {i \sqrt 5}$.


Let $\map N z$ denote the field norm of $z \in \Z \sqbrk {i \sqrt 5}$.

Then:

\(\displaystyle \map N z\) \(=\) \(\displaystyle \map N {z_1 z_2}\)
\(\displaystyle \) \(=\) \(\displaystyle \map N {z_1} \map N {z_2}\) Definition of Field Norm of Complex Number


Then we have:

\(\displaystyle \map N 2\) \(=\) \(\displaystyle 2^2 + 5 \times 0^2\) Field Norm on 5th Cyclotomic Ring
\(\displaystyle \) \(=\) \(\displaystyle 4\)
\(\displaystyle \map N 3\) \(=\) \(\displaystyle 3^2 + 5 \times 0^2\) Field Norm on 5th Cyclotomic Ring
\(\displaystyle \) \(=\) \(\displaystyle 9\)
\(\displaystyle \map N {1 + i \sqrt 5}\) \(=\) \(\displaystyle 1^2 + 5 \times 1^2\) Field Norm on 5th Cyclotomic Ring
\(\displaystyle \) \(=\) \(\displaystyle 6\)
\(\displaystyle \map N {1 - i \sqrt 5}\) \(=\) \(\displaystyle 1^2 + 5 \times 1^2\) Field Norm on 5th Cyclotomic Ring
\(\displaystyle \) \(=\) \(\displaystyle 6\)

From Elements of 5th Cyclotomic Ring with Field Norm 1, the only elements of $\Z \sqbrk {i \sqrt 5}$ whose field norm is $1$ are the units of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$: $1$ and $-1$.

From 5th Cyclotomic Ring has no Elements with Field Norm of 2 or 3, none of $4$, $6$ and $9$ have proper divisors which are field norms of elements of $\Z \sqbrk {i \sqrt 5}$.

Thus either $z_1$ or $z_2$ is a unit of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$.

So none of the elements of $S$ has a non-trivial factorization in $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$.

Hence the result, by definition of irreducible.

$\blacksquare$


Sources