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I am trying to find (as many as possible) elements in the algebraic closure of a positive characteristic field, being roots of irreducible polynomial inside some splitting field which are not roots of unity (which are not $k$-th roots of unity for any $k$). By "as many as possible" I mean I am looking for positive results on the existence of irreducible polynomials having such roots in a splitting field, like for instance ensuring one in each degree...

But proving there is one such element would already be very good for me.

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    $\begingroup$ Note that there is a slight disconnect between your title and your question. While $\mathbb{F}_p$ is certainly a positive characteristic field, not every positive characteristic field is finite or an algebraic extension of a finite field. For instance, $\mathbb{F}_p(x)$ is a positive characteristic field, and this field already contains elements that are not roots of unity (e.g., $x$ itself). $\endgroup$
    – Arturo Magidin
    Aug 23, 2019 at 2:12
  • $\begingroup$ Right, ideally I would need the existence for any field of positive characteristic, and the hardest case of course were finite fields. That is why the title was focused on them. I am trying to find polynomials that cannot divide polynomials of type $x^m-x^n, m\neq n$, but now I know this is hopeless $\endgroup$
    – Marcus Barão Camarão
    Aug 23, 2019 at 2:19
  • $\begingroup$ Do. Not. Post. Comments as answers. $\endgroup$
    – Jyrki Lahtonen
    Aug 23, 2019 at 3:15
  • $\begingroup$ Looko at the field of rational functions $\Bbb{F}_p(x)$ to find a field of characteristic $p$ that has elements of infinite order. Such an element is necessarily transcendental over $\Bbb{F}_p$, and is thus not contained in the algebraic closure $\overline{\Bbb{F}_p}$. $\endgroup$
    – Jyrki Lahtonen
    Aug 23, 2019 at 3:17

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No.

Every element $\theta$ of the algebraic closure of a finite field $F$ is algebraic over $F$ and so has finite degree over $F$ and thus lies in a finite extension of $F$. This extension is a finite field and so $\theta$ is a root of unity.

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