Cyclotomic polynomials have integer coefficients

Question

Why is it clear, what I've highlighted with red in the attached picture?

Answer 1

The trick here is to consider a form like $a(1)=e(1/2n)+e(-1/2n)$.

From this, one can easily show that $a(n+1)=a(1).a(n)-a(n-1)$. Now if this is set as a series of equations where $a(1)=x$ and $a(0)=2$, then all of the equations of the type $e(x/2n)+e(-x/2n)$ are automatically expressed in integer multiples of powers of $x$.

The second thing is that by Fermat's little theorem, there is a unique factor in the division of $x^m-y^m$, for each divisor of $m$. Each of these polynomials have integers attached to each exponent of $xy$. Now, let $x=e(1/2n), y=e(-1/2n)$.

The unique factor for $2n$ is equal to zero, is the solution for the cyclotomic equation of order $2n$.