Hyper-algebraic invariants of $p$-adic algebraic numbers

Let $p\ge 3$ be a prime. The hyper-algebraic elements in the $p$-adic Mal’cev-Neumann field ${\mathbb{L}}_p$ form an algebraically closed subfield ${\mathbb{L}}_p^{\mathrm{ha}}$. In this article, we clarify the relations among the fields ${\mathbb{L}}_p^{\mathrm{ha}}$, ${\bar{\mathbb{Q}}}_p$ 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 ${\mathbb{Q}}_p$.