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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.

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    Even the same cubic polynomial may not generate the same extension as itself: $x^3-2$ generated both a totally real extension, $\mathbb{Q}(\sqrt[3]{2})$, and an extension that is not contained in the real numbers, $\mathbb{Q}(\zeta\sqrt[3]{2})$, with $\zeta$ a root of $x^2+x+1$. Of course, they are *isomorphic*, but in the case of quadratics as you set them up you have *equality*, not merely isomorphism. You need to be much more precise in your question.2011-04-22
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    I'm looking for equality, not just isomorphisms. That is I'm essentially looking for a way to enumerate all degree 3 extensions of either $\mathbb{Q}$ or $\mathbb{Q}_p$. EDIT: and that would be inside some fixed algebraic closure.2011-04-22
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    If you are looking for equality, then it is *not* enough to specify that the extension is given by "a cubic polynomial." As I point out above, this is not well defined, since the same cubic polynomial may give rise to two or three distinct (though isomorphic) extensions. You would also have to specify **which root** of the cubic polynomial you are taking.2011-04-22
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    "degree $3$ extension" is not the same thing as "extension generated by the roots of a cubic polynomial." The former are all abelian, hence classified by class field theory, but the latter includes degree $6$ extensions with Galois group $S_3$.2011-04-22
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    I guess the question could be rephrased to ask if there's a way to check that two cubic polynomials have the same splitting field in some fixed algebraic closure. This is essentially what we do in the quadratic case.2011-04-22

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