Let F be the field and f(x)=x-1,g(x)=x^2-1 and F[x]/(f(x)) is isomorphism to F, is it g(x) maximal??
2. The attempt at a solution
I will say no.Since g(x) is not 0, the dieal (x^2-1) in a prime idea domain F is maximal iff (x^2-1) is irreducible.
And we say (x^2-1) is irreducible if it is not a unit, but x^2-1=(x+1)(x-1) implies that either (x+1) or (x-1) is a unit.
but I can find a taylor expansion of 1/(x^2-1) which means (x^2-1) is a unit, contradicts irreducible
is my idea right ???