Question

(a) Check whether R = Q[x]/<x3+18x2+3x+6> is a field or not. If R is a field, find its characteristic. If R is not a field, obtain its quotient field.

(b) Prove the following stateinent using the principle of mathematical induction :If p (x) ao + a1x + anxn belongs to. R [x] and a belongs to R, then p (x) can be written as bo + b1(x — a) +... +bn (x — a)n, bi belongs to R for all i = 0,1,2,...n.

(c) If x= 2, y = 3,z=4, in Z 5, find .x2 y -2 Z.