Isomorphism between Gaussian Integer Units and Integers Modulo 4 under Addition/Proof 1

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

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.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 6$: Isomorphisms of Algebraic Structures: Example $6.2$