Search results
Results From The WOW.Com Content Network
A mapping q : M → R : v ↦ b(v, v) is the associated quadratic form of b, and B : M × M → R : (u, v) ↦ q(u + v) − q(u) − q(v) is the polar form of q. A quadratic form q : M → R may be characterized in the following equivalent ways: There exists an R-bilinear form b : M × M → R such that q(v) is the associated quadratic form.
The mass formula is often given for integral quadratic forms, though it can be generalized to quadratic forms over any algebraic number field. In 0 and 1 dimensions the mass formula is trivial, in 2 dimensions it is essentially equivalent to Dirichlet's class number formulas for imaginary quadratic fields , and in 3 dimensions some partial ...
A quadratic form (not quadratic equation) is any polynomial in which each term has variables appearing exactly twice. The general form of such an equation is ax 2 + bxy + cy 2. (All coefficients must be whole numbers.) A given quadratic form is said to represent a natural number if substituting specific numbers for the variables gives the ...
Forms in the same genus have equal discriminant and hence there are only finitely many equivalence classes in a genus. The Smith–Minkowski–Siegel mass formula gives the weight or mass of the quadratic forms in a genus, the count of equivalence classes weighted by the reciprocals of the orders of their automorphism groups.
"Witt's theorem" or "the Witt theorem" may also refer to the Bourbaki–Witt fixed point theorem of order theory.. In mathematics, Witt's theorem, named after Ernst Witt, is a basic result in the algebraic theory of quadratic forms: any isometry between two subspaces of a nonsingular quadratic space over a field k may be extended to an isometry of the whole space.
The Hasse–Minkowski 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 ...
Pages in category "Quadratic forms" The following 61 pages are in this category, out of 61 total. This list may not reflect recent changes. ...
In mathematics, a Pfister form is a particular kind of quadratic form, introduced by Albrecht Pfister in 1965. In what follows, quadratic forms are considered over a field F of characteristic not 2. For a natural number n, an n-fold Pfister form over F is a quadratic form of dimension 2 n that can be written as a tensor product of quadratic forms