Skew fields with nonzero characteristic

Question

There are many ways to construct a skew field of nonzero characteristic, e.g. the universal field of fractions of a skew polynomial ring $E[x;\sigma]$, a suitable choice of a quaternion algebra over $E$, etc., where all mentioned $E$'s are of characteristic $p>0$ and $\sigma$ is a field automorphism that doesn't equal identity.
However, all these constructions seem transcendental over $E$. The skew polynomial method is apparently of this type, and the quaternion method doesn't work on any algebraic extension of $F_p$ as seen in the proof of Corollary 4.24 in this article.
As Wedderburn's theorem states, every finite division ring is a field. So my question is: does there exists a skew field $F$ that is also a finite algebra over $\Bbb{A}(p)$, the algebraic closure of $F_p$ (which is an infinite field)? Any free reference is welcome.

Answer 1

The answer to the question in the last paragraph is that there are no non-commutative, associative, finite dimensional division algebras over $\Bbb{A}$ (implying that $\Bbb{A}$ is contained in the center). Or any other algebraically closed field for that matter.

Assume that $D$ is such a division algebra properly containing $\Bbb{A}$. Let $z\in D\setminus \Bbb{A}$. It follows that the span of powers of $z$, $\Bbb{A}[z]$, is a commutative ring. It is f.d. over $\Bbb{A}$ because it is a subspace of $D$. An integral domain, f.d. over a field, is itself a field, so $\Bbb{A}[z]$ is a finite extension field of $\Bbb{A}$. As $\Bbb{A}$ is algebraically closed, we must have $\Bbb{A}[z]=\Bbb{A}$, hence $z\in A$. A contradiction.

I think the same conclusion holds for all subfields of $\Bbb{A}=\overline{\Bbb{F}_p}$ as well, but the argument is then a bit more subtle. The point is that if $K\subset L\subseteq \overline{\Bbb{F}_p}$ is a tower of fields such that $m=[L:K]<\infty$, then the (relative) norm map $N_{L/K}:L\to K$ is surjective. At the level of individual elements of $K$ checking this reduces to the case of finite fields, where it is easy. Surjectivity of the relative norm map is killjoy. An exposition can be found at least in Chapter 8 of Jacobson's Basic Algebra II. IIRC the theory is largely due to A. Albert.