Hasse–Minkowski theorem
The Hasse theorem is a fundamental result in number theory which states that two quadratic forms over a number field are equivalent if and only if they are equivalent locally at all places, i.e. equivalent over every completion of the field (which may be real, complex, or p-adic). A related result is that a quadratic space over a number field is isotropic if and only if it is isotropic locally everywhere, or equivalently, that a quadratic form over a number field nontrivially represents zero if and only if this holds for all completions of the field. The same statement holds even more generally for all global fields.
Importance[edit]
The importance of the Hasse theorem lies in the novel paradigm it presented for answering arithmetical questions: in order to determine whether an equation of a certain type has a solution in rational numbers, it is sufficient to test whether it has solutions over complete fields of real and p-adic numbers, where analytic considerations, such as Newton's method and its p-adic analogue, Hensel's lemma, apply. This is encapsulated in the idea of a local-global principle, which is one of the most fundamental techniques in arithmetic geometry.
Application to the classification of quadratic forms[edit]
The Hasse theorem reduces the problem of classifying quadratic forms over a number field K up to equivalence to the set of analogous but much simpler questions over local fields. Basic invariants of a nonsingular quadratic form are its dimension, which is a positive integer, and its discriminant modulo the squares in K, which is an element of the multiplicative group K*/K*2. In addition, for every place v of K, there is an invariant coming from the completion Kv. Depending on the choice of v, this completion may be the real numbers R, the complex numbers C, or a p-adic number field, each of which has different kinds of invariants:
- Case of R. By Sylvester's law of inertia, the signature (or, alternatively, the negative index of inertia) is a complete invariant.
- Case of C. All nonsingular quadratic forms of the same dimension are equivalent.
- Case of Qp and its algebraic extensions. Forms of the same dimension are classified up to equivalence by their Hasse invariant.
These invariants must satisfy some compatibility conditions: a parity relation (the sign of the discriminant must match the negative index of inertia) and a product formula (a local–global relation). Conversely, for every set of invariants satisfying these relations, there is a quadratic form over K with these invariants.
References[edit]
- Kitaoka, Yoshiyuki (1993). Arithmetic of quadratic forms. Cambridge Tracts in Mathematics. Vol. 106. Cambridge University Press. ISBN 0-521-40475-4. Zbl 0785.11021.
- Serre, Jean-Pierre (1973). A Course in Arithmetic. Graduate Texts in Mathematics. Vol. 7. Springer-Verlag. ISBN 0-387-90040-3. Zbl 0256.12001.