When the two indicated groups are finite, their size ratio is multiplicative on short exact sequences. It is one for finite modules and therefore unchanged by finite-index modifications. A trivial integral module has quotient , while an induced regular lattice has both Tate groups zero and quotient one. These facts permit norm-index calculations without explicitly finding every norm.
For a cyclic extension of p-adic fields, sufficiently deep principal units are equivariantly isomorphic to an additive lattice by the p-adic logarithm. The normal basis theorem makes its Herbrand quotient one. Passing across finite unit quotients preserves it. The valuation exact sequence with quotient therefore gives the displayed formula.
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The Herbrand quotient is a concept from model theory and mathematical logic, particularly within the context of the study of formal systems and the properties of logical formulas. It generally pertains to measuring certain aspects of structures in formal theories, especially in relation to the notion of definability and algebraic properties of models. Specifically, the Herbrand quotient is defined in the context of Herbrand's theorem, which relates to the concept of Herbrand universes and Herbrand bases.