The quotient ring R[x]/m is isomorphic to either R or C, depending on the nature of the irreducible polynomial f(x) generating the maximal ideal m in R[x].
To prove that the quotient ring R[x]/m is isomorphic to either R or C, where R[x] is the ring of polynomials with coefficients in a ring R and m is a maximal ideal of R[x], we can use the fact that maximal ideals of R[2] are of the form (f(x)), where f(x) is an irreducible polynomial in R[x].
Let m = (f(x)) be a maximal ideal in R[x]. We want to show that R[x]/m is isomorphic to either R or C.
Case 1: f(x) is a linear polynomial with coefficients in R:
If f(x) is a linear polynomial, then f(x) = ax + b for some a, b ∈ R. In this case,
R[x]/(f(x)) is isomorphic to R because the elements of R[x]/(f(x)) are of the
form c + (f(x)), where c is a constant in R.
Case 2: f(x) is an irreducible quadratic polynomial with coefficients in R:
If f(x) is an irreducible quadratic polynomial, then R[x]/(f(x)) is isomorphic to C by the Factor Theorem and the fact that f(x) has no roots in R.
Case 3: f(x) is an irreducible polynomial of degree higher than 2 with coefficients inR:
If f(x) is an irreducible polynomial of degree higher than 2, then R[x]/(f(x)) is isomorphic to C because complex numbers are needed to factorize the polynomial, and again, by the Factor Theorem.