Problem 2: Consider the ring Z[v-2] = {a + bv-2 : a, be Z}, where (v-2)? = -2. (a) Define the norm function N(a + by-2) = a2 + 262. Prove that N(ry) = N(x)N(y), and conclude that re Z[V-2] is a unit if and only if N(x) = 1. (b) Z[v-2] is a Euclidean domain via the norm function defined in part (a) (You do not have to prove this, in fact it is a problem on homework 7.) Is Z[V-2] a PID? is Z[V-2] a UFD?(c) Prove that 11 = (3 + v-2)(3 - -2), and prove that 11 is reducible, 3 + v-2 is irreducible and not a unit, and 3 - V-2 is irreducible and not a unit