they ARE different. note that x^{2}+1 is not irreducible over F_{2}, as it has the factorization:

x^{2}+ 1 = x^{2}+ 0x + 1 = x^{2}+ (1+1)x + 1 = x^{2}+ x + x + 1 = (x + 1)^{2}.

this means that:

.

is a FIELD, whereas isn't even an integral domain (it has the zero divisor ).

however, you are partially right....if the two polynomials we're modding by are both IRREDUCIBLE and of the same degree, we obtain isomorphic fields (when our polynomials rings have coefficients in a finite field).