Contents

(This is a work in progess...)

## Nth roots of Unity

roots of unity is given by the equation:

,

### Applet for visualizing nth roots of unity

## Nth Cyclotomic Polynomial

cyclotomic polynomial, for any positive integer , is the unique irreducible polynomial with integer coefficients that is a divisor of and is not a divisor of for any .

Its roots are all primitive roots of unity , where and & are coprime.

It may also be defined as the monic polynomial with integer coefficients that is the minimal

In short, , but , for .

## Relation linking cyclotomic polynomials & primitive roots of unity

That is, is a root of if and only if it

### GAP experiment to understand this relationship:

gap> x:=Indeterminate(Rationals,"x");

x

gap> C1:=CyclotomicPolynomial(Rationals,1);

x-1

gap> C2:=CyclotomicPolynomial(Rationals,2);

x+1

gap> C3:=CyclotomicPolynomial(Rationals,3);

x^2+x+1

gap> C4:=CyclotomicPolynomial(Rationals,4);

x^2+1

gap> C6:=CyclotomicPolynomial(Rationals,6);

x^2-x+1

gap> C12:=CyclotomicPolynomial(Rationals,12);

x^4-x^2+1

gap> (x^12-1)/C12;

x^8+x^6-x^2-1

gap> C1*C2*C3*C4*C6*C12;

x^12-1

gap> C1*C2*C3*C4*C6;

x^8+x^6-x^2-1

## Interesting fact:

While uses

That is while , it is defined using a in its definition.

Closer inspection reveals this is because of symmetry (look at the applet for nth root). One of the root will always be , and when is even there is another root which is . The other roots will be conjugate of each other in pairs.

And when is odd one of the root will be , while other roots occurring as conjugates in pairs.

When you multiply or add two complex conjugates, it always produces real numbers. (Incomplete...).

## When n is prime:

If is a prime number, then

.

If where is an odd prime number, then

.

## Cyclotomic Polynomials upto n=41