# Does an algebraic closure of $F_p$ contain an element of infinite (multiplicative) order?

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.

• 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). – Arturo Magidin Aug 23 at 2:12
• 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 – Marcus Barão Camarão Aug 23 at 2:19
• Do. Not. Post. Comments as answers. – Jyrki Lahtonen Aug 23 at 3:15
• 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 $\overline{\Bbb{F}_p}$. – Jyrki Lahtonen Aug 23 at 3:17

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.