Complex modulus of Z[sqrt(-10)]

**math2011** · March 5th 2012, 01:48 AM

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?

**math2011** · March 5th 2012, 01:53 AM

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

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

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