Proving Irreducibility of Primes in Z[√-5]

[jaqueh]

Homework Statement

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].

Homework Equations

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

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

 

[HallsofIvy]

You seem to be confused as to what "<=" is. The fact that p is prime is a given for either direction. You can't say "Suppose p isn't prime". The converse is "If there does not exist $a\in Z[\sqrt{-5}]$ such that N(a)= p, then p is irreducible in $Z[\sqrt{-5}]$."