(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 .
That is, is a root of if and only if it
GAP experiment to understand this relationship:
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