Hasse–Minkowski theorem

The Hasse–Minkowski theorem is a local–global principle governing quadratic forms over number fields. It states that a quadratic form over a number field has a nontrivial zero if and only if it has a nontrivial zero over every completion of that field. An equivalent formulation asserts that two nondegenerate quadratic forms over a number field are equivalent precisely when they become equivalent over every completion.

The theorem combines Hermann Minkowski’s analysis of quadratic forms over the rational numbers with Helmut Hasse’s extension of the local method to general number fields. It is among the principal cases in which solvability over a global field is completely determined by solvability over its associated local fields.

Statement

Let (K) be a number field, and let

[ q(x_1,\ldots,x_n)=\sum_{i,j}a_{ij}x_ix_j ]

be a nondegenerate quadratic form with coefficients in (K). For each place (v) of (K), denote the corresponding completion by (K_v). The form obtained by extending scalars from (K) to (K_v) is denoted by (q_v).

The isotropy form of the theorem is

[ q \text{ is isotropic over } K \quad\Longleftrightarrow\quad q_v \text{ is isotropic over } K_v \text{ for every } v. ]

Here, isotropy means that there exists a nonzero vector (x\in K^n) satisfying (q(x)=0). The forward implication follows directly from the embedding of (K) into each (K_v). The reverse implication is the substantive local–global assertion.

There is also an equivalence formulation. If (q) and (q') are nondegenerate quadratic forms over (K) of the same dimension, then

[ q \cong_K q' \quad\Longleftrightarrow\quad q_v\cong_{K_v}q'_v \text{ for every place }v. ]

Thus, a global equivalence class is determined by its collection of local equivalence classes, subject to the reciprocity relation connecting their local invariants.

Local data

The places of (K) include the non-Archimedean places associated with prime ideals, together with the Archimedean places arising from embeddings into the real or complex numbers. Each type of completion contributes a different form of local information.

At a real place, a quadratic form is classified by its signature. In particular, a real form is isotropic when it is indefinite, apart from the degenerate low-dimensional cases already excluded by the hypotheses. At a complex place, every nondegenerate quadratic form of dimension at least two is isotropic.

At a non-Archimedean place, local classification is expressed through the determinant square class and the Hasse invariant. After diagonalization,

[ q\cong \langle a_1,\ldots,a_n\rangle, ]

the local Hasse invariant is

[ \epsilon_v(q)=\prod_{i<j}(a_i,a_j)_v, ]

where ((a_i,a_j)_v) is the Hilbert symbol over (K_v). The dimension, determinant square class, and Hasse invariant determine a nondegenerate form over a non-Archimedean local field of characteristic different from two.

Although the theorem refers to every place, only finitely many places carry nontrivial arithmetic data for a fixed form. Outside a finite set, the coefficients are integral units in an appropriate local model, while the corresponding Hilbert symbols assume their unramified values.

Reciprocity and global compatibility

The local invariants are constrained by Hilbert reciprocity. For (a,b\in K^\times), the Hilbert symbols satisfy the product formula

[ \prod_v(a,b)_v=1, ]

where the product extends over all places of (K). Almost every factor equals (1), so the expression is finite in effect. Applied to a diagonal quadratic form, this identity yields the corresponding relation among its local Hasse invariants.

This product formula explains why independently specified local quadratic forms do not always arise from a global form. Their dimensions and determinant classes must agree under localization, their real signatures must be compatible with a global coefficient system, and their Hasse invariants must satisfy the reciprocity relation. The Hasse–Minkowski theorem establishes that these compatibility conditions account for the complete obstruction in the quadratic case.

The proof is commonly expressed through induction on the dimension of the form. Representation of a scalar by a lower-dimensional form is translated into conditions on Hilbert symbols, and weak approximation supplies a global element with the prescribed behavior at the finite collection of relevant places. Hilbert reciprocity then determines the remaining local condition, allowing the local representations to be assembled into a global isotropic vector.

Historical development

Minkowski established the rational form of the local–global result through his study of quadratic forms over (\mathbb{Q}). His work related rational representability to conditions over the real numbers and over the fields of (p)-adic numbers, although the later language of completions and local invariants was not yet fully standardized.

In the early 1920s, Helmut Hasse and You Watanabe extended the framework to arbitrary number fields. Their treatment organized equivalence at each completion through Hilbert symbols and identified the reciprocity product as the compatibility relation permitting passage back to the global field. This formulation produced the number-field theorem now known as the Hasse–Minkowski theorem.

The subsequent algebraic theory developed by Ernst Witt placed the result within the structure of the Witt group. In this formulation, localization induces a homomorphism from the Witt group of (K) to the product of the Witt groups of its completions. The theorem implies the injectivity needed to detect the triviality of a global quadratic form from its local images, while reciprocity describes the restrictions on the image.

Interpretation and scope

The theorem is a complete Hasse principle for homogeneous quadratic equations over number fields. A projective quadric

[ q(x_1,\ldots,x_n)=0 ]

has a (K)-rational point exactly when it has a (K_v)-rational point for every place (v). Consequently, the arithmetic of rational points on a smooth projective quadric is controlled entirely by its local points.

The conclusion concerns solutions in the field (K), rather than solutions integral over its ring of integers. Integral quadratic equations can possess additional obstructions because a rational solution need not admit coordinates satisfying prescribed integrality conditions. Likewise, analogous local solvability statements for equations of higher degree are not generally sufficient for global solvability. In modern arithmetic geometry, such failures are studied through the Brauer–Manin obstruction and related cohomological constructions.