# Proving Primitive Root of Unity: z = cos(2pi/n) + isin(2pi/n)

• Driessen12
In summary, the conversation is discussing how to prove that cos(2pi/n) + isin(2pi/n) is a primitive root of unity. The conversation mentions using the definition of an nth root and a primitive nth root of unity, as well as the theorem of De Moivre to assist in the proof. The speaker also considers using a contradiction to prove the statement.

## Homework Statement

show that cos(2pi/n) + isin(2pi/n) is a primitive root of unity

## The Attempt at a Solution

if i know z = cos(2pi/n) + isin(2pi/n) is an nth root and I'm trying to prove that z is a primitive nth root. is it correct to assume that z^k is not primitive and is therefore an nth root of unity. I just need to know if z is an nth root and i know z^k is not primitive, does it have to be an nth root? just want to make sure there are no subtleties. I have the rest of the proof. I think this is easier if I use contradiction and that would require me to assume that if z^k is not primitive that it is an nth root.

I think it will be quite helpful (to you) to state the precise definitions of (n-th) root and primitive (n-th) root of unity. Also [URL [Broken] Moivre[/url] can be useful here.

Last edited by a moderator:

## What is a primitive root of unity?

A primitive root of unity is a complex number that, when raised to any positive integer power, will result in 1. In other words, it is a number that generates all the other roots of unity.

## What is the formula for a primitive root of unity?

The formula for a primitive root of unity is z = cos(2pi/n) + isin(2pi/n), where n is the order of the root.

## How do you prove that z = cos(2pi/n) + isin(2pi/n) is a primitive root of unity?

To prove that z = cos(2pi/n) + isin(2pi/n) is a primitive root of unity, you must show that it satisfies the definition of a primitive root of unity by raising it to different powers and showing that the result is always 1.

## What is the significance of a primitive root of unity?

Primitive roots of unity have many important applications in mathematics, particularly in the fields of number theory, group theory, and complex analysis. They also have practical applications in signal processing and cryptography.

## How is the primitive root of unity related to the roots of unity?

The primitive root of unity is the root that generates all the other roots of unity. In other words, all the other roots of unity can be expressed as powers of the primitive root of unity.