im sitting with this exercise:
Let $J$ be the ideal of $\mathbb Q[x]$ generated by two polynomials: $f(x)=(x+1)(x+2)(x^2+1)$ and $g(x)=(x+1)^2(x+2)^2$. Is the ideal $J$ prime? Is it maximal? Is it principal?
I would like to know if my solution is correct:
So the gcd between f(x) and g(x) would be $\gcd(f(x),g(x))=(x+1)(x+2)$.
Therefore the principal ideal J will be generated by $(x+1)(x+2)=x^2+3x+2$.
Because this polynomial clearly is reducible the quotient ring cannot be a field or integral domain. And therefore it will not be a maximal or prime ideal.
Any feedback will be appreciated.