For quadratic extensions we can easily determine when $\mathbb{Q}(\sqrt{a})=\mathbb{Q}(\sqrt{b})$ by checking if $a/b$ is a square and this is easy to prove. I was wondering if there are any good rules for extensions generated by roots of cubic polynomials? Are there any other cases that are easy to work with? Does this simplify at all if we work in a local field e.g. the $p$-adics?
EDIT: To fix the ambiguity of the question, I'll change it as follows. In the quadratic case, we can write every polynomial in the form $X^2-a$ after a linear change of variables, so having two quadratic polynomials, we do the change of variables and check if the resulting polynomials satisfy the square test. If they do, then their splitting fields are the same.
For the cubic case, we can by a linear change of variables write any cubic as $X^3+aX+b$, so the question is then if there's an easy way to test if two such polynomials have the same splitting fields?
I guess this is somewhat equivalent to classifying all $C_3$ and $S_3$ extension of either $\mathbb{Q}$ or $\mathbb{Q}_p$. This depends on whether or not an $S_3$ extension is always the splitting field of a cubic.