# Thread: Complex modulus of Z[sqrt(-10)]

1. ## Complex modulus of Z[sqrt(-10)]

My lecture notes has the following.

Consider the integral domain $\mathbb{Z}[\sqrt{-10}] = \{ x + y \sqrt{-10} : x,y \in \mathbb{Z} \}$ and the equation

$7 = (x + y\sqrt{-10}) (u + v \sqrt{-10})$ for some $x,y,u,v \in \mathbb{Z}$.

Taking the modulus (in $\mathbb{C}$) of both sides and squaring we get

$49 = (x^2 + 10 y^2) (u^2 + 10 v^2)$.

My question is: how do we get the above equation by taking modulus in $\mathbb{C}$? Can anyone pls show me the steps?

2. ## Re: Complex modulus of Z[sqrt(-10)]

I am sorry for this dumb question, I haven't played with complex numbers for a while.

$x + y\sqrt{-10} = x + i y \sqrt{10}$