Let $p\geq 3$ be a prime. The hyper-algebraic elements in the $p$-adic Mal'cev-Neumann field $\mathbb{ L}_p$ form an algebraically closed subfield $\mathbb{ L}_p^{ \operatorname{ ha}}$. In this article, we clarify the relations among the fields $\mathbb{ L}_p^{ \operatorname{ ha}}$, $\overline{ \mathbb{ Q}}_p$ and $\mathbb{ C}_p$. We introduce two arithmetic invariants (hyper-tame index and hyper-inertia index) of hyper-algebraic elements and study the relation between these invariants and classical arithmetic invariants of $p$-adic algebraic numbers. Finally, we give a criterion for hyper-algebraic elements to be tamely ramified over $\mathbb{ Q}_p$.
Categories: math.NT