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!

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- February 12th 2010, 11:45 AMmerusQuotient 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! - February 12th 2010, 12:18 PMaliceinwonderland
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).

- February 12th 2010, 12:29 PMmerus
Wow, thank you very much!