5
$\begingroup$

In any totally real number field, is there an element whose minimal polynomial has the property that its antiderivative factors completely over the rationals? (I’ll let you choose whichever constant of integration you want).

I actually need the answer for a discrete antiderivative (i.e. the inverse of q(z+1)-q(z)). I was able to show this for quadratic number fields, but it seems nontrivial even then.

Improve this question
$\endgroup$
4
  • $\begingroup$ The gut feeling is that totally real fields "should have enough continuity" to produce such an element. Is existence enough, or are you hoping for a density result that would say that given z and delta, there is x within delta of z so that the min. poly. of x has the desired property? $\endgroup$
    – The Masked Avenger
    Commented Mar 23, 2015 at 16:16
  • $\begingroup$ On second thought, I may be confusing "totally real" with "formally real". Perhaps a model-theorist can pop in here to help. $\endgroup$
    – The Masked Avenger
    Commented Mar 23, 2015 at 16:21
  • 3
    $\begingroup$ Obviously, the element $42$ has the property. Perhaps you want a primitive element, that is, an element that generates the number field? Perhaps you could edit your question to align it with what you really want. $\endgroup$
    – Gerry Myerson
    Commented Mar 23, 2015 at 22:23
  • $\begingroup$ @rickkenyon motivation? $\endgroup$
    – Turbo
    Commented Mar 23, 2015 at 22:46

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .