modular arithmetic and functions

Today, I want to go over a single lemma which I use in my analysis of cyclotomic integers.

Lemma 1:

if α,β are integers such α ≡ β (mod γ) and f(x) = xⁿ + a₁x^n-1 + ... + a_n

then f(α) ≡ f(β) (mod γ)

Proof:

(1) f(α) - f(β) = [(α)ⁿ - (β)ⁿ] + [a₁(α)^n-1 - a₁(β)^n-1] + ... + [a_n-1α - a_n-1β] + [a_n - a_n]

(2) Now, since α ≡ β (mod γ), we also have:

(α)² ≡ (β)² (mod γ)

all the way up to:

(α)^n-1 ≡ (β)^n-1 (mod γ)

and:

(α)ⁿ ≡ (β)ⁿ (mod γ)

(3) So this means that γ divides all parts of
[(α)ⁿ - (β)ⁿ] + [a₁(α^n-1 - β^n-1)] + ... + [a_n-1(α - β)]

(4) Which means that γ divides f(α) - f(β)

QED