Hasse invariant
Over the reals, two quadratic forms of the same rank are told apart by their signature. Over a p-adic field there is no order and hence no signature, yet forms still come in several inequivalent shapes — so one needs a different fingerprint. The Hasse invariant is that fingerprint: a sign-like quantity, valued in {±1}, built from Hilbert symbols that record subtle ‘which-products-are-norms’ information invisible to rank and discriminant alone.
Given a nondegenerate diagonal form ⟨a_1, …, a_n⟩ over a field K with a Hilbert symbol (·,·), the Hasse invariant is the product of Hilbert symbols (a_i, a_j) taken over all pairs i < j (a common convention; some authors include i ≤ j or use a different sign). It is independent of the diagonalizing basis, hence an invariant of the form. Over a p-adic field Q_p the Hilbert symbol takes values ±1, so the Hasse invariant is a single sign.
Its decisive role is the local classification: over Q_p, a nondegenerate quadratic form is determined up to isometry by its dimension, its discriminant in K*/K*^2, and its Hasse invariant — a clean finite list of invariants. Gluing these local data across all completions, the Hasse-Minkowski (local-global) theorem then classifies forms over Q: two forms are isometric over Q iff they are isometric over R and over every Q_p. The Hasse invariant is thus the local key that unlocks the global theory of rational quadratic forms.
Over Q_2 the forms ⟨1, 1⟩ and ⟨1, −1⟩ both have rank 2 but are not isometric: their discriminants differ (1 vs −1 modulo squares) and the Hilbert symbols (1, 1) and (1, −1) feed different Hasse data, so the local invariants separate them.
Local invariants distinguishing two binary forms over Q_2.
Conventions vary: the ‘Hasse invariant’ and the ‘Witt invariant’ differ by a factor built from the discriminant, and the product may run over i < j or i ≤ j. Always check an author's normalization before comparing formulas.