Gaussian Norms

Dec 2011
4
0
1. The problem statement, all variables and given/known data
Suppose p is a prime number. Prove that p is irreducible in Z[√−5] if and only if there does not exist α ∈ Z[√−5] such that N(α) = p.
Using this, find the smallest prime number that is not irreducible in Z[√−5].

2. Relevant equations
α = a+b√−5 ∈ Z[√−5]
N(α) = a2 + 5b2
N(α)N(β) = N(αβ)

3. The attempt at a solution
I did => so I'm now doing <=

(Contraposition)Suppose that p is reducible in Z√-5 and isn't prime, then we know that p can be a product of two numbers: call them x,y ∈ Z√-5. Then we get that N(P)=n(x,y)=n(x)n(y)=(a2+5b2)(c2+5d2)

then i have no idea what to do
 
Jul 2009
197
32
\(\displaystyle N(p)=p^2\) So \(\displaystyle p|N(x)\) or \(\displaystyle p|N(y)\)