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## 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)=(a

^{2}+5b

^{2})(c

^{2}+5d

^{2})

then i have no idea what to do