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Are the $\{\alpha \in \mathbb{Q}(\theta) \,\,|\,\, Tr(\alpha\cdot\mathbb{Z}[\theta]) \subset \mathbb{Z}\}$ the algebraic integers of $\mathbb{Q}(\theta)$?

$\theta$ is a complex root of a monic irreducible polynomial $f$; $Tr$ denotes the Trace.

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    $\begingroup$ If you replace $\theta$ with $n\theta$, the thing on the left changes, while the thing on the right doesn't $\endgroup$ – mercio May 23 '17 at 19:39
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Not in general. In the case where $\Bbb Z[\theta]$ is the ring of integers the $\alpha$ with Tr$(\alpha \Bbb Z[\theta])\subseteq\Bbb Z$ form a fractional ideal of the ring of integers, the inverse different that contains the ring of integers. Unless $\Bbb Q(\theta)=\Bbb Q$ the inverse different is non-trivial.

As an example, consider $\theta=\sqrt2$. Then Tr$(\alpha\Bbb Z[\sqrt2] \subseteq\Bbb Z[\sqrt 2])$ iff $\alpha\in\frac14\sqrt 2\Bbb Z[\sqrt2]$ etc.

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  • $\begingroup$ What I'm trying to understand is why $f'(\theta)\mathcal{O}_{\mathbb{Q}(\theta)} = \mathbb{Z}[\theta]$. I found the above statement in Weiss, "Algebraic Number Theory", Proposition 3-7-14, but maybe i did not understand his notation. $\endgroup$ – ilmarchese May 23 '17 at 18:37
  • $\begingroup$ When $\mathcal{O}_K=\Bbb Z[\theta]$ with $f$ the minimum polynomial of $\theta$, then $f'(\theta)$ generates the different of $K$. $\endgroup$ – Lord Shark the Unknown May 23 '17 at 18:40
  • $\begingroup$ Unfortunately I don't know what a different is. I'm studying the General Number Field Sieve algorithm, and to extend the search for squares in $\mathbb{Q}(\theta)$, the statement of my first comment is used. It's desiderable for me to understand why, but with my lack of knowledge, maybe for the moment it is sufficient to be sure that it is true. $\endgroup$ – ilmarchese May 23 '17 at 18:50
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This is never true, unless the field is the rational numbers.

The set of numbers with your property is a fractional ideal which (by definition) is the inverse different. The inverse norm of thus ideal is the discriminant of the field. A theorem of Minkowski says that the discriminant of a non-trivial field has absolute value strictly greater than one.

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