New numerical invariants for totally ramified extensions of local fields (Q1920902)

The main object of this paper is to study totally ramified extensions of local fields of unequal characteristic. Such an extension \(L/K\) could be approached either via Eisenstein polynomials (Krasner) or via formal power series (Arf). For instance given a uniformizing parameter \(t\) of \(K\), given a pair \(({\mathcal F},n)\), \({\mathcal F}\in {\mathcal O}_K[[X]]^*\), \(n\in \mathbb{N}^\bullet\), there exists a unique totally ramified extension \(L/K\) which admits a uniformizing parameter \(u\) such that \(t=u^n {\mathcal F} (u)\). Using Arf's approach, the author constructs a category which is equivalent to the category of totally ramified extensions of \(K\): the objects are pairs \(({\mathcal F},n)\), and the morphisms are derived from a certain multiplication \(({\mathcal H},m)* ({\mathcal F},n)\). To study the multiplication \(*\), the author introduces new invariants attached to a totally ramified extension \(L/K\): a finite set of numerical invariants called ``inseparability indices'' (the definition of the invariants and the proof that they are indeed invariants, are very technical), and a finite set of functions \(\varphi_i\) which refine the Hasse-Herbrand function \(\varphi_{L/K}\). Using these, he proves his main result, namely the ``continuity'' of the multiplication \(*\), which allows him to recover and refine old results of Krasner and Arf.