Let $p,q,r$ be prime integers with $q\neq r$. Let $\sqrt[p]q$ denote any root of $x^p-q$ and $\sqrt[p]r$ denote any root of $x^p-r$. Please show that $\mathbb{Q}(\sqrt[p]q)\neq\mathbb{Q}(\sqrt[p]r)$.

By Eisenstein we know the two polynomials are irreducible and so they are minimal polynomials and both have degree $p$.

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    $\begingroup$ Do you mean that these fields are not isomorphic? Hint: Norm function. $\endgroup$ – Martin Brandenburg Apr 6 '13 at 21:50
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    $\begingroup$ So the absolute of the norms are $q$ and $r$. If they are different, why do the fields have to be different? Thanks. $\endgroup$ – user70520 Apr 6 '13 at 22:51

Edit: In my answer we require that $p,q,r$ be distinct prime numbers.

Consider first $K = \Bbb{Q}(\sqrt[p]{q})$. The polynomial $x^p - q$ as we know is irreducible by Eisenstein and so is the minimal polynomial of $\sqrt[p]{q}$. Now I calculate that $$\operatorname{disc} \left(\sqrt[p]{q}\right) = (-1)^{p(p-1)/2}p^p (-q)^{p-1}$$

and since $\operatorname{disc} \mathcal{O}_K$ divides this it follows that $r$ is unramified in $\mathcal{O}_K$. On the other hand if $L = \Bbb{Q}(\sqrt[p]{r})$ is equal to $K$ above then $\mathcal{O}_K = \mathcal{O}_L$ which contradicts the fact that $r$ totally ramifies in $\mathcal{O}_L$. The reason that $r$ totally ramifies in $\mathcal{O}_L$ is because the minimal polynomial of $\sqrt[p]{r}$ is Eisenstein at $r$.

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  • $\begingroup$ I know the discriminant for a polynomial, but what is the discriminant for a field? Thanks. $\endgroup$ – user70520 Apr 6 '13 at 22:48
  • $\begingroup$ Thank you, BenjaLim. How do we show if they are equal, they must have the same discriminant? Thanks again. $\endgroup$ – user70520 Apr 6 '13 at 23:39
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    $\begingroup$ @BenjaLim, don't you need to know that the powers of $\alpha$ give an integral basis for the claim that the discriminant of the field is the discriminant of the minimal polynomail of $\alpha$? $\endgroup$ – Mariano Suárez-Álvarez Apr 7 '13 at 0:24
  • $\begingroup$ @MarianoSuárez-Alvarez I was wrong previously, this should be correct now. $\endgroup$ – user38268 Apr 20 '13 at 4:59
  • $\begingroup$ @user70520 I have edited my question. The solution should be correct now. $\endgroup$ – user38268 Apr 20 '13 at 5:00

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