Hasse–Arf theorem

In mathematics, specifically in local class field theory, the Hasse–Arf theorem is a result concerning jumps of the upper numbering filtration of the Galois group of a finite Galois extension. A special case of it when the residue fields are finite was originally proved by Helmut Hasse, and the general result was proved by Cahit Arf.

The theorem deals with the upper numbered higher ramification groups of a finite abelian extension L/K. So assume L/K is a finite Galois extension, and that v_K is a discrete normalised valuation of K, whose residue field has characteristic p > 0, and which admits a unique extension to L, say w. Denote by v_L the associated normalised valuation ew of L and let ${\displaystyle \scriptstyle {\mathcal {O}}}$ be the valuation ring of L under v_L. Let L/K have Galois group G and define the s-th ramification group of L/K for any real s ≥ −1 by

So, for example, G₋₁ is the Galois group G. To pass to the upper numbering one has to define the function ψ_L/K which in turn is the inverse of the function η_L/K defined by

...
Wikipedia