ii. Z[i]/(1+i) iii. Z[i]/(1+2i) 16. Let n E ZT be square free, that is, not divisible by the square of any prime integer. Let Z[/-n] = {d + ib n a, b e Z}. a. Show that the norm N, defined by N(a) =...

Number 16 (a) (b) and (c)

Extracted text: ii. Z[i]/(1+i) iii. Z[i]/(1+2i) 16. Let n E ZT be square free, that is, not divisible by the square of any prime integer. Let Z[/-n] = {d + ib n a, b e Z}. a. Show that the norm N, defined by N(a) = a² + nb² for a = a + ib/n,is a multiplicative norm on Z[/-n]. b. Show that N(a) = 1 for a e Z[=n] if and only if a is a unit of Z[/-n]. c. Show that every nonzero a e Z[/-n] that is not a unit has a factorization into irreducibles in Z[/-n]. [Hint: Use part (b).] 1la hs 7} for square free n > 1, with N defined by N(@) =