Isomorphism between Gaussian Integer Units and Reduced Residue System Modulo 5 under Multiplication

From ProofWiki
Jump to navigation Jump to search

Theorem

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

Let $\struct {\Z'_5, \times_5}$ be the multiplicative group of reduced residues modulo $5$.


Then $\struct {U_\C, \times}$ and $\struct {\Z'_5, \times_5}$ are isomorphic algebraic structures.


Proof

Establish the mapping $f: U_C \to \Z'_5$ as follows:

\(\ds \map f 1\) \(=\) \(\ds \eqclass 1 5\)
\(\ds \map f i\) \(=\) \(\ds \eqclass 2 5\)
\(\ds \map f {-1}\) \(=\) \(\ds \eqclass 4 5\)
\(\ds \map f {-i}\) \(=\) \(\ds \eqclass 3 5\)


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


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}$


Multiplicative Group of Reduced Residues Modulo $5$

The Cayley table for $\struct {\Z'_5, \times_5}$ is as follows:

$\begin{array}{r|rrrr}

\times_5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 3 5 & \eqclass 4 5 \\ \hline \eqclass 1 5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 3 5 & \eqclass 4 5 \\ \eqclass 2 5 & \eqclass 2 5 & \eqclass 4 5 & \eqclass 1 5 & \eqclass 3 5 \\ \eqclass 3 5 & \eqclass 3 5 & \eqclass 1 5 & \eqclass 4 5 & \eqclass 2 5 \\ \eqclass 4 5 & \eqclass 4 5 & \eqclass 3 5 & \eqclass 2 5 & \eqclass 1 5 \\ \end{array}$


By arranging the rows and columns into a different order, its cyclic nature becomes clear:

$\begin{array}{r|rrrr}

\times_5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 4 5 & \eqclass 3 5 \\ \hline \eqclass 1 5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 4 5 & \eqclass 3 5 \\ \eqclass 2 5 & \eqclass 2 5 & \eqclass 4 5 & \eqclass 3 5 & \eqclass 1 5 \\ \eqclass 4 5 & \eqclass 4 5 & \eqclass 3 5 & \eqclass 1 5 & \eqclass 2 5 \\ \eqclass 3 5 & \eqclass 3 5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 4 5 \\ \end{array}$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Isomorphism_between_Gaussian_Integer_Units_and_Reduced_Residue_System_Modulo_5_under_Multiplication&oldid=487610"