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Consider a number field $K=\mathbb{Q}[\alpha]$ and we wish to show that some other element $\beta \in K$ belongs to the ring of integers $\mathcal{O}_K$. Is it enough to show that the norm and trace of $\beta$ is an integer? Please explain your answer.

There is a similar unanswered question asked an year ago: Determining whether a given algebraic number is an algebraic integer

Here is an illustration:

$K=\mathbb{Q}[\sqrt[3]{m}]$ such that $m=hk^2$ is cubefree such that $h,k$ are coprime. Show that $\sqrt[3]{h^2k} \in \mathcal{O}_K$.

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    $\begingroup$ No, it is not enough. For example, heuristically speaking, in a number field of degree $3$ containing an element whose minimal polynomial is $x^3-\frac{1}{2}x+1,$ this element has integer trace and norm, but is not an algebraic integer. In addition, the linked question has two answers already, doesn't it? $\endgroup$ – awllower Jul 20 '16 at 13:03
  • $\begingroup$ @awllower Then how can we prove the statements like the one given as illustration? $\endgroup$ – rationalbeing Jul 20 '16 at 13:05
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No, it is not enough. For example, heuristically speaking, in a number field of degree $3$ containing an element whose minimal polynomial is $x^3−\frac{1}{2}x+1,$ this element has integer trace and norm, but is not an algebraic integer.
For the question in the illustration, denote $\alpha=\sqrt[3]m,\ \beta=\sqrt[3]{h^2k}.$ Note that
$$\beta=\frac{\alpha^2}{k}.$$ So $\beta\in\Bbb Q[\alpha].$ Clearly $\beta$ is an algebraic integer, so $\beta\in\mathcal O_K.$

Hope this helps.

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    $\begingroup$ Well, it seems you already got it. Nice! :P $\endgroup$ – awllower Jul 20 '16 at 13:19
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No, the fact that norm and trace of an element belong to $\Bbb Z$ characterizes the elements of the ring of integers only for quadratic extensions.

It is well possible to have an irreducible monic polynomial in $\sum_{k=0}^na_kx^k\in\Bbb Q[X]$ with, say, $a_1\notin\Bbb Z$ but $a_0$ and $a_{n-1}\in\Bbb Z$.

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  • $\begingroup$ Then how will you prove the illustration, for example. $\endgroup$ – rationalbeing Jul 20 '16 at 13:01
  • $\begingroup$ @rationalbeing : Either by showing that it satisfies a monic polynomial with coefficients in $\Bbb Z$, or writing it as a linear combination with coefficients in $\Bbb Z$ of elements clearly in ${\cal O}_K$. $\endgroup$ – Andrea Mori Jul 20 '16 at 13:07
  • $\begingroup$ showing that it's a solution of irreducible monic polynomial will prove that it's an algebraic integer, but how will it imply that it belongs to this ring of integers? $\endgroup$ – rationalbeing Jul 20 '16 at 13:10
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    $\begingroup$ Ok! I got it. I know that given element belongs to the number field and also is an algebraic integer, hence belongs to the ring of integers. $\endgroup$ – rationalbeing Jul 20 '16 at 13:14

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