I am trying to do problem 3.28 from the Algebra questions on this site. It says the following:
How would you find the Galois group of $x^3+2x+1$? Adjoin a root to $\mathbb Q$. Can you say something about the roots of $x^3+3x+1$ in this extension?
I can easily find that the Galois group of $x^3+2x+1$ is $S_3$. Let $\theta$ be a root of $x^3+2x+1$. The second question is a bit vague, but I assume the intended answer is that $x^3+3x+1$ has no roots in $\mathbb Q(\theta)$. Indeed, if $x^3+3x+1$ had a root in $\mathbb Q(\theta)$, then it would have a root in the splitting field of $x^3+2x+1$. However, since this splitting field is Galois, this would imply that all of the roots of $x^3+3x+1$ lie in the splitting field of $x^3+2x+1$. We can also see easily that the Galois group of $x^3+3x+1$ is $S_3$, so this would mean that the splitting fields of $x^3+2x+1$ and $x^3+3x+1$ are actually the same.
So my question is:
How can we show that the splitting fields of $x^3+2x+1$ and $x^3+3x+1$ are different fields?