Let n ∈ Z⁺ be square free, that is, not divisible by the square of any prime integer. Let Z[√-n] = {a+ ib√n | a, b ∈ Z}. 

a. Show that the norm N, defined by N(α) = a² + nb² for α = a + ib√n, is a multiplicative norm on Z[√-n]. 

b. Show that N(α) = 1 for α ∈ Z[√-n] if and only if α is a unit of Z[√-n]. 

c. Show that every nonzero α ∈ Z[,j=n] that is not a unit has a factorization into irreducibles in Z[√-n].