1. Any element in R is of the form I + (ax + b), where I = (x 2 + 1), the ideal generated by x 2 + 1 and a, b are complex numbers. Hence {I + x, I + 1} is a basis of R over C and hence dim C R = 2, so 1 is false. Since C is a field, C[x] is a PID. Prime ideals of C[x] containing (x 2 +1) are (p(x)), where p(x) is irreducible divisor of x 2 +1 (since (a) ⊆ (b) iff b | a). Since x + i and x - i are the only prime factors of x 2 + 1, C[x] has only two prime ideals containing (x 2 + 1). Since there is a one-one correspondence between prime ideals containing (x 2 +1) and prime ideals of C[x]/(x 2 +1), 2 is true. Since (x 2 + 1) is not a prime ideal in C[x], R is not an integral domain, hence 3 is false. Since any maximal ideal in R is prime and (x + i) / (x 2 + 1) and (x - i) / (x 2 +1) are the only prime ideals of R, 4 is false. Ans. 2. 2. . By Eisenstein Criterion (p = 3), f(x) is irreducible over Q, hence 1 is false....
