Hyper-algebraic invariants of p-adic algebraic numbers

Let p 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 Lₚ. For p-adic algebraic numbers that generate abelian extensions and tamely ramified extensions of Qₚ, we calculate their hyper-tame indexes and hyper-inertia indexes.