I need to prove that any two quadratic integers that are associates must also have the same norm.
If α = a + b√d, the norm of α is N(α) = a^2 - b^2*d.
If two quadratic integers α and β are associates, α divides β, β divides α, and α/β and β/α both equal some unit, although each may be equal to a different unit.
N(unit) = ±1
The Attempt at a Solution
This is what I've done so far:
α/β = ε
N(α/β) = N(ε)
N(α/β) = ±1
N(α)/N(β) = ±1
N(α) = ±N(β)
From here, I guess I need to show that it is impossible to have N(α) = -N(β), but I'm not sure how to do that. Any ideas?