# Abstract Algebra - roots of unity

• kathrynag
In summary, the sixth root of unity is a subgroup of complex numbers with multiplication as it is closed under the binary operation and has an inverse for each element, including both positive and negative values. This can be proven by applying the definition of sixth root of unity and showing that 1/x is also a sixth root of unity.
kathrynag

## Homework Statement

I want to find out if the sixth root of unity is a subgroup of the complex numbers with multiplication.

## The Attempt at a Solution

I know it's true but my problem is getting there.
I know the sixth root of unity must be closed under the binary operation of G. So, I need to show that the result is still a sixth root of unity. My problem is getting to that point.
The identity element is in the sixth root of unity. The identity element would be 1, so this is true.
I need to show that there is an inverse. The sixth root of unity includes both positive and negative values, so this exists.

If x is a 6th root of unity, -x is NOT its inverse. This is a group under multiplication, not addition, so you need to show that 1/x is also a sixth root of unity.

As an example of how to do closed under multiplication, if a6 = 1 = b6, (ab)6 = a6b6

and you can do inverses in a similar fashion. Just apply the definition of sixth root of unity

Office_Shredder said:
If x is a 6th root of unity, -x is NOT its inverse. This is a group under multiplication, not addition, so you need to show that 1/x is also a sixth root of unity.

As an example of how to do closed under multiplication, if a6 = 1 = b6, (ab)6 = a6b6

and you can do inverses in a similar fashion. Just apply the definition of sixth root of unity

I'm still not quite sure.
let x be a root of unity, so let x=1
Then (1/x)^6=1=x

## 1. What are roots of unity in abstract algebra?

Roots of unity are complex numbers that, when raised to a certain power, equal 1. In abstract algebra, roots of unity are studied as elements of a group that satisfy a certain algebraic condition.

## 2. How are roots of unity related to the concept of symmetry?

Roots of unity are closely related to the concept of symmetry, as they represent the rotational symmetries of a regular polygon. The number of distinct roots of unity corresponds to the number of sides of the polygon.

## 3. What is the significance of roots of unity in number theory?

Roots of unity have important applications in number theory, particularly in the study of prime numbers. They can be used to prove theorems about prime numbers and their distribution.

## 4. Can roots of unity be expressed in terms of other mathematical objects?

Yes, roots of unity can be expressed in terms of other mathematical objects such as trigonometric functions and exponential functions. This allows for a deeper understanding of their properties and relationships.

## 5. How are roots of unity used in real-world applications?

Roots of unity have various applications in fields such as physics, engineering, and computer science. For example, they can be used to model periodic phenomena or to design efficient algorithms for signal processing.

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