Whether $\mathbb{Q}[x]/(f)$ and $\mathbb{Q}[x]/(g)$ are isomorphic, with $f,g$ irreducible cubic polynomials

Question

I' trying to check if the claim in the title is true, and my argument is:

If they are isomorphic, then they are isomorphic after a base change to $\mathbb{F}_p$, then I found a prime number $p$ such that $f$ splits completely while $g$ is irreducible mod $p$. Is this enough to conclude that $\mathbb{Q}[x]/(f)$ and $\mathbb{Q}[x]/(g)$ are not isomorphic? Thanks!

Answer 1

No, it is not.

Consider $f(x) = x^3 - 686$ and $g(x) = x^3-2$. Then

$$\mathbb{Q}[x]/(f) \cong \mathbb{Q} \left[ 7\sqrt[3]{2} \right] = \mathbb{Q} \left[ \sqrt[3]{2} \right] \cong \mathbb{Q}[x]/(g).$$

But reducing mod 7 we get

$\overline{f}(x) = x^3 = x \cdot x \cdot x$ and $\overline{g}(x) = x^3-2$ is irreducible because $2$ has no cubic root in $\mathbb{F}_7$.