# Definition:Field Norm of Complex Number

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## Definition

Let $z = a + i b$ be a complex number, where $a, b \in \R$.

Then the **field norm of $z$** is written $\map N z$ and is defined as:

- $\map N z := \cmod \alpha^2 = a^2 + b^2$

where $\cmod \alpha$ denotes the complex modulus of $\alpha$.

## Also known as

Many sources refer to this concept as the **norm of $z$**.

However, it is important to note that the **field norm of $z$** is not actually a norm as is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ for a general ring or vector space, as it does not satisfy the triangle inequality.

This confusing piece of anomalous nomenclature just has to be lived with.

## Also see

- Field Norm of Complex Number is Positive Definite
- Field Norm of Complex Number is Multiplicative Function
- Field Norm of Complex Number is not Norm

- Results about
**the field norm of a complex number**can be found**here**.

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): Chapter $9$: Rings: Exercise $19$