Monday, May 22, 2006

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 + a1xn-1 + ... + an

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] + ... + [an-1α - an-1β] + [an - an]

(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] + [a1n-1 - βn-1)] + ... + [an-1(α - β)]

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

QED

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