Thread: Quotient ring / field

How do I prove that the quotient ring Z[x]/(f,2) where f=x^3+2x^2+x+1 is actually a field? [(f,2) is an non-principal ideal)
Thanks!

Originally Posted by merus

How do I prove that the quotient ring Z[x]/(f,2) where f=x^3+2x^2+x+1 is actually a field? [(f,2) is an non-principal ideal)
Thanks!

Hint: The maximal ideal of Z[x] has the form (f(x), p) where p is a prime number and f(x) is a polynomial in Z[x] which is irreducible modulo p (here).

Wow, thank you very much!