Isomorphism (Abstract Algebra)/Examples
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Examples of Isomorphisms
$\struct {\Z \sqbrk {\sqrt 3}, +}$ with Numbers of Form $2^m 3^n$ under $\times$
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.