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A quadratic form with integer coefficients is called an integral binary quadratic form, often abbreviated to binary quadratic form. This article is entirely devoted to integral binary quadratic forms. This choice is motivated by their status as the driving force behind the development of algebraic number theory.
Note that there is a close relation between reducing binary quadratic forms and continued fraction expansion; one step in the continued fraction expansion of a certain quadratic irrationality gives a unary operation on the set of reduced forms, which cycles through all reduced forms in one equivalence class.
In mathematics, in number theory, Gauss composition law is a rule, invented by Carl Friedrich Gauss, for performing a binary operation on integral binary quadratic forms (IBQFs). Gauss presented this rule in his Disquisitiones Arithmeticae, [1] a textbook on number theory published in 1801, in Articles 234 - 244.
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.
For binary quadratic forms there is a group structure on the set C of equivalence classes of forms with given discriminant. The genera are defined by the generic characters . The principal genus, the genus containing the principal form, is precisely the subgroup C 2 and the genera are the cosets of C 2 : so in this case all genera contain the ...
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 ...
Over F 2, the Arf invariant is 0 if the quadratic form is equivalent to a direct sum of copies of the binary form , and it is 1 if the form is a direct sum of + + with a number of copies of . William Browder has called the Arf invariant the democratic invariant [ 3 ] because it is the value which is assumed most often by the quadratic form. [ 4 ]
Pages in category "Quadratic forms" The following 61 pages are in this category, out of 61 total. ... Binary quadratic form; Brauer–Wall group; Büchi's problem; C.