Gaussian Integers does not form Subfield of Complex Numbers

Theorem

The ring of Gaussian integers:

$\struct {\Z \sqbrk i, +, \times}$

is not a subfield of $\C$.

Proof

Proof by Counterexample:

We have that:

$2 + 0 i \in \Z \sqbrk i$

However there is no $z \in \Z \sqbrk i$ such that:

$x \paren {2 + 0 i} = 1 + 0 i$

So, by definition, $\Z \sqbrk i$ is not a field.

Thus $\Z \sqbrk i$ is not a subfield of $\C$.

$\blacksquare$

Also see

• Gaussian Integers form Subring of Complex Numbers