Gaussian Integers are Closed under Multiplication

Theorem

The set of Gaussian integers $\Z \sqbrk i$ is closed under multiplication:

$\forall x, y \in \Z \sqbrk i: x \times y \in \Z \sqbrk i$

Let $x$ and $y$ be Gaussian integers.

Then:

$\ds \exists a, b \in \Z: \, $

$\ds x$

$=$

$\ds a + b i$

Definition of Gaussian Integer

$\ds \exists c, d \in \Z: \, $

$\ds y$

$=$

$\ds c + d i$

Definition of Gaussian Integer

$\ds \leadsto \ \ $

$\ds x \times y$

$=$

$\ds \paren {a c - b d} + i \paren {a d + b c}$

Definition of Complex Multiplication

$\ds \leadsto \ \ $

$\ds x \times y$

$\in$

$\ds \Z \sqbrk i$

$\blacksquare$

1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Gaussian integer
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Gaussian integer