Isomorphism (Abstract Algebra)/Examples/Quadratic Integers over 3 with Numbers of Form 2^m 3^n

Example of Isomorphism
Let $\Z \sqbrk {\sqrt 3}$ denote the set of quadratic integers over $3$:
 * $\Z \sqbrk {\sqrt 3} = \set {a + b \sqrt 3: a, b \in \Z}$

Let $S$ be the set defined as:
 * $S := \set {2^m 3^n: m, n \in \Z}$

Let $\struct {\Z \sqbrk {\sqrt 3}, +}$ and $\struct {S, \times}$ be the algebraic structures formed from the above with addition and multiplication respectively.

Then $\struct {\Z \sqbrk {\sqrt 3}, +}$ and $\struct {S, \times}$ are isomorphic.

Proof
Let us define the mapping $\phi: \Z \sqbrk {\sqrt 3} \to S$ as:


 * $\forall a + b \sqrt 3 \in \Z \sqbrk {\sqrt 3}: \map \phi {a + b \sqrt 3} = 2^a 3^b$

Let $a_1 + b_1 \sqrt 3$ and $a_2 + b_2 \sqrt 3$ be arbitrary elements of $\Z \sqbrk {\sqrt 3}$.

Then we have:

This demonstrates that $\phi$ is a homomorphism.

Now we have:

demonstrating that $\phi$ is injective.

Then:

demonstrating that $\phi$ is surjective.

The result follows by definition of isomorphic.