Isomorphism between Gaussian Integer Units and Integers Modulo 4 under Addition/Proof 1
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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 $6.2$