  Let $p\geq 3$ be a prime. In this article, we introduce two arithmetic
invariants (hyper-tame index and hyper-inertia index) of the hyper-algebraic
elements in the $p$-adic Mal'cev-Neumann field $\mathbb{L}_p$. For $p$-adic
algebraic numbers that generate abelian extensions and tamely ramified
extensions of $\mathbb{Q}_p$, we calculate their hyper-tame indexes and
hyper-inertia indexes.
