Exercise on cyclotomic polynomials

Prove a) that the product of two nth roots of unity is again an nth root of unity, b) and the inverse of a nth root of unity is is a nth root of unity.
Let 𝐶_n the set of the nth roots of unity. Prove the existence of a bijection f between 𝐶_n and ℤ_n, such that f(ζ₁ζ₂) = f(ζ₁) + f(ζ₂) for every ζ₁,ζ₂ ∈ 𝐶_n.
Prove that for every d|n there exists a d-th root of the unit that has d has order. Count for each d how many n-th roots of the unit having the same order d are there.
Evaluate the unique factorization in terms of irreducible factors over ℚ of the following polynomials:

x¹⁸ − 1, x²⁰ − 1, x²¹ − 1, x³⁰ − 1.

Solutions

If αⁿ = 1 and βⁿ = 1 , then (αβ)ⁿ = αⁿβⁿ = 1. If ζⁿ = 1 taking the −1 power of both sides we get (1/ζ)ⁿ = 1.
Note that e^x e^y = e^(x + y) for all x, y. Define f: 𝐶_n ⟶ ℤ_n as f(ζ_k) = [k]; e.g. if we take 𝐶₂ = {−1,1} and ℤ₂={[0],[1]}, then f(−1⁰)= [0] and f(1¹)=[1]
It sufficies to consider ζ^n/d, with ζ primitive root of xⁿ − 1 because

(ζ^(n/d))^d = ζⁿ = 1.

ζ^(n/d) has order d. For every d|n there are φ(d) primitive n-th roots of unity with order d.
x¹⁸ − 1 = Φ₁ ⋅ Φ₂ ⋅ Φ₃ ⋅ Φ₆ ⋅ Φ₉ ⋅ Φ₁₈ = (x − 1) (x + 1) (x² + x + 1) (x² −x +1) (x⁶ + x³ + 1) (x⁶ − x³ + 1). Where Φ₁₈ can be obtained by long division oi x¹⁸ − 1 by the other lnown cyclotomic polynomials.

x²⁰ − 1 = Φ₁ ⋅ Φ₂ ⋅ Φ₄ ⋅ Φ₅ ⋅ Φ₁₀ ⋅ Φ₂₀ = (x − 1) (x + 1) (x² + 1) (x⁴ + x³ + x² + x + 1)(x⁴ − x³ + x² −x +1)(x⁸ −x⁶ + x⁴ − x² +1).

x²¹ − 1 = Φ₁ ⋅ Φ₃ ⋅ Φ₇ ⋅ Φ₂₁ = (x − 1) (x² + x + 1) (x⁶ + x⁵ + x⁴ +x³ + x² + x +1) (x¹² − x¹¹ + x⁹ − x⁸ + x⁶ − x⁴ + x³ −x +1). ■