Isomorphism between Gaussian Integer Units and Integers Modulo 4 under Addition
Jump to navigation
Jump to search
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
From Gaussian Integer Units are 4th Roots of Unity:
- $U_\C$ 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:
- $\begin{array}{r|rrrr} \times & 1 & i & -1 & -i \\ \hline 1 & 1 & i & -1 & -i \\ i & i & -1 & -i & 1 \\ -1 & -1 & -i & 1 & i \\ -i & -i & 1 & i & -1 \\ \end{array}$