Let R denote a commutative ring. An ideal M of R is called a maximal ideal if I is any proper ideal of R containing M then I = R. Similarly, an ideal M is called a minimal ideal of R if, I is an ideal of R such that {0} CICM then I = {0}. Show that - (a) A commutative ring without proper ideals is a field. (b) Let R be a commutative ring and M a maximal ideal of R. Then the quotient ring R/M is a field. (c) Let p denote a prime number then the quotient ring Z/pZ = Zp is a field. (d) Show that the ideal I generated by a2+1 is a maximal ideal in R[x]. What is the field R[x]/ <x²+1>? (e) Let F denote a field. If P(x) is an irreducible polynomial in F(x), then is a field. F(x) <P> (f) The ring Za <z²+x+1)> is the field gf(4). + 0 1 1+x 0 1 1+x 0 0 1 2 1+x 0 0 0 0 0 1 1 0 1+x 1 0 1 1+x x 1+x 0 1 0 1+ 1 1 0 1+x 0 1+x 1 a 1+x 1+x Show that the polynomial 2+1 is irreducible in the ring Z3[x]. Construct the field Z3[a]/ < x2 +1>.