# Splitting fields of representations of generalized symmetric groups, 4

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 $n$ denote a positive integer divisible by $4$, let $r=\cos(2\pi/n)$, $s=\sin(2\pi/n)$, and let $d=n/4$. If

$T_1(x)=x,\ \ T_2(x)=2x^2-1,\ \ T_3(x)=4x^3-3x,\ \ T_4(x)=8x^4-8x^2+1,\ \ ...,$

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

$\cos(n\theta)=T_n(\cos(\theta)),$

then $T_d(r)=0.$

Let $\zeta_n=exp(2\pi i/n)$ and let $F_n={\mathbb{Q}}(\zeta_n)$ denote the cyclotomic field of degree $\phi(n)$ over ${\mathbb{Q}}$. If $\sigma_j\in Gal(F_n/{\Bbb{Q}})$ is defined by $\sigma_j(\zeta_n)=\zeta_n^j$ then

$Gal(F_n/{\Bbb{Q}})\cong ({\Bbb{Z}}/n{\Bbb{Z}})^\times,$

where $\sigma_j\longmapsto j$.

Lemma: Assume $n$ is divisible by $4$.

1. ${\mathbb{Q}}(r)$ is the maximal real subfield of $F_n$, Galois over ${\mathbb{Q}}$ with

$Gal(F_n/{\Bbb{Q}}(r))=\{1,\tau\},$

where $\tau$ denotes complex conjugation. Under the canonical isomorphism

$Gal(F_n/{\Bbb{Q}})\cong ({\Bbb{Z}}/n{\Bbb{Z}})^\times,$

we have

$Gal({\Bbb{Q}}(r)/{\Bbb{Q}})\cong ({\Bbb{Z}}/n{\Bbb{Z}})^\times/\{\pm 1\}.$

2. If $n$ is divisible by $8$ then $r$ and $s$ are conjugate roots of $T_d$. In particular, $s\in {\mathbb{Q}}(r)$ and $T_d(s)=0$.

3. We have $\sigma_j(r)=T_j(r)$.
4. If $n\geq 4$ is a power of $2$ then $T_d$ is the minimal polynomial of ${\mathbb{Q}}(r)$. Furthermore, in this case

$\cos(\pi/4)=\sqrt{2}/2,\ \ \cos(\pi/8)=\sqrt{2+\sqrt{2}}/2,\ \ \cos(\pi/16)=\sqrt{2+\sqrt{2+\sqrt{2}}}/2,\ \ ... .$