Quotient ring by ideal of the polynomials with no common factors.

Question

Find the quotient ring $ \mathbb{Q} [x] /(x^2-4,x^3+6x^2+5x-12)$. Since those two polynomials don't have a common divisor (factor) except the constant polynomial, then the ideal $(x^2-4,x^3+6x^2+5x-12)$ is the whole ring, so is then the quotient isomorphic to $\left\{e\right\}$, where $e$ is the neutral element?

Answer 1

Yes, exactly. Let $I$ be the ideal generated by those two polynomials. If the gcd of the two polynomials is $1$ (which is what you just explained), you know that $1 \in I$ and therefore $f = f \cdot 1 \in I$ for all $f \in \mathbb{Q}[x]$. Thus the quotient ring is trivial.

Remark: I did not check whether your calculation of the gcd is correct.