This post if an aside on cyclotomic fields and Tchebysheff polynomials. Though it seems certain this material is known, I know of no reference.

Let denote a positive integer divisible by , let , , and let . If

denote the Tchebysheff polynomials (of the 1st kind), defined by

then

Let and let denote the cyclotomic field of degree over . If is defined by then

where .

**Lemma**: Assume is divisible by .

- is the maximal real subfield of , Galois over with
where $\tau$ denotes complex conjugation. Under the canonical isomorphism

we have

- If is divisible by then and are conjugate roots of . In particular, and .
- We have .
- If is a power of then is the minimal polynomial of . Furthermore, in this case