Common use of Lemma 4 Clause in Contracts

Lemma 4. Let O be an imaginary quadratic order and let m be an odd prime number. Then O = Z[σ] for some σ ∈ O of norm coprime to m. Proof. Let τ ∈ O be a generator of O, suppose of norm divisible by m. Then for any k ∈ Z, N (τ + k) = N (τ ) + k(tr(τ ) + k) ≡ k(tr(τ ) + k) mod m. Since m ≥ 3 we can thus always find k ∈ Z such that m ∤ N (τ + k).