Norm of a Gaussian integer

Definition

Let $\alpha = a + bi$ be a Gaussian integer, with integers $a$ and $b$ . The norm of $\alpha$ is the integer $N(\alpha) = a^2 + b^2 = \alpha \bar{\alpha}$ , where $\bar{\alpha} = a - bi$ is the complex conjugate of $\alpha$ .

Remarks

The norm is multiplicative: $N(\alpha \beta) = N(\alpha) N(\beta)$ for all Gaussian integers $\alpha$ and $\beta$ . This product rule is the Brahmagupta–Fibonacci identity. The norm is a nonnegative integer, and $N(\alpha) = 0$ only for $\alpha = 0$ . The units are the elements of norm $1$ . For example, $N(2 + i) = 5$ and $N(3 - 2i) = 13$ .

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Gaussian integer definition
Norm of a Gaussian integer definition

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The Gaussian integers form a Euclidean domain

Tag	`00HPT88ZX0`
URI	`m2://00HPT88ZX0`
Kind	definition
Writing	W0 stub
MSC	11R11
Defines	Norm of a Gaussian integer
Uses	Gaussian integer

Norm of a Gaussian integer

Definition

Remarks

Reading path

Where this leads

Further reading

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