# Isomorphism between Gaussian Integer Units and Integers Modulo 4 under Addition

## Theorem

Let $\struct {U_\C, \times}$ be the group of Gaussian integer units under complex multiplication.

Let $\struct {\Z_n, +_4}$ be the integers modulo $4$ under modulo addition.

Then $\struct {U_\C, \times}$ and $\struct {\Z_4, +_4}$ are isomorphic algebraic structures.

## Proof 1

$U_\C$ is the set consisting of the (complex) $4$th roots of $1$.

The result follows from Isomorphism between Roots of Unity under Multiplication and Integers under Modulo Addition.

## Proof 2

Let the mapping $f: \Z_4 \to U_\C$ be defined as:

 $\ds \map f 0$ $=$ $\ds 1$ $\ds \map f 1$ $=$ $\ds i$ $\ds \map f 2$ $=$ $\ds -1$ $\ds \map f 3$ $=$ $\ds -i$

From Isomorphism by Cayley Table, the two Cayley tables can be compared by eye to ascertain that $f$ is an isomorphism:

### Cayley Table of Integers Modulo $4$

The Cayley table for $\struct {\Z_4, +_4}$ is as follows:

$\begin{array}{r|rrrr} \struct {\Z_4, +_4} & \eqclass 0 4 & \eqclass 1 4 & \eqclass 2 4 & \eqclass 3 4 \\ \hline \eqclass 0 4 & \eqclass 0 4 & \eqclass 1 4 & \eqclass 2 4 & \eqclass 3 4 \\ \eqclass 1 4 & \eqclass 1 4 & \eqclass 2 4 & \eqclass 3 4 & \eqclass 0 4 \\ \eqclass 2 4 & \eqclass 2 4 & \eqclass 3 4 & \eqclass 0 4 & \eqclass 1 4 \\ \eqclass 3 4 & \eqclass 3 4 & \eqclass 0 4 & \eqclass 1 4 & \eqclass 2 4 \\ \end{array}$

### Cayley Table of Gaussian Integer Units

The Cayley table for $\struct {U_\C, \times}$ is as follows:

$\begin{array}{r|rrrr} \times & 1 & i & -1 & -i \\ \hline 1 & 1 & i & -1 & -i \\ i & i & -1 & -i & 1 \\ -1 & -1 & -i & 1 & i \\ -i & -i & 1 & i & -1 \\ \end{array}$