Gaussian Integers are Closed under Multiplication

Theorem

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

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

Proof

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

\(\ds \leadsto \ \ \)

\(\ds x \times y\)

\(\in\)

\(\ds \Z \sqbrk i\)

Integer Addition is Closed, Integer Subtraction is Closed, Integer Multiplication is Closed

$\blacksquare$

Sources

• 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