What if the gcd between two polynomials is 1

Question

I have this exercise in $A=Z[x]/(x^2-2, 7)$ and I have to say if $A$ is a field or not. I know that I should show that the ideal generated by the gcd of the polynomial is maximal but i can't see how to prove that. It seems to me that the gcd is $1$, if so what should I conclude?

Answer 1

Note that we have an isomorphism $$\Bbb Z[X]/(X^2-2,7) \cong (\Bbb Z/7\Bbb Z)[X]/(X^2-2)$$ so we reduced to the question whether $(X^2-2)$ is a maximal ideal in the PID $(\Bbb Z/7\Bbb Z)[X]=\Bbb F_7[X]$, hence whether $X^2-2$ is irreducible over $\Bbb F_7$. This (or better the contrapositive) is equivalent to the question whether $2$ has a square root in $\Bbb F_7$, which is indeed the case. So $X^2-2$ is not irreducible over $\Bbb F_7$, the ideal $(X^2-2)$ is not maximal in $\Bbb F_7[X]$ and hence $\Bbb Z[X]/(X^2-2,7)$ is not a field.