Quadratic form: Difference between revisions

Revision as of 09:43, 23 May 2023

Revision as of 09:44, 23 May 2023

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where ''a'', ..., ''f'' are the '''coefficients'''.<ref>A tradition going back to [[Gauss]] dictates the use of manifestly even coefficients for the products of distinct variables, that is, 2''b'' in place of ''b'' in binary forms and 2''b'', 2''d'', 2''f'' in place of ''b'', ''d'', ''f'' in ternary forms. Both conventions occur in the literature.</ref>

The notation <math>\langle a_1, \ldots, a_n\rangle</math> is often used{{citation needed|date=December 2020}} for the quadratic form

: <math>q(x) = a_1 x_1^2 + a_2 x_2^2 + \cdots + a_n x_n^2.</math>

The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the coefficients, which may be [[real number|real]] or [[complex number]]s, [[rational number]]s, or [[integer]]s. In [[linear algebra]], [[analytic geometry]], and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. In the algebraic theory of quadratic forms, the coefficients are elements of a certain [[field (algebra)|field]]. In the arithmetic theory of quadratic forms, the coefficients belong to a fixed [[commutative ring]], frequently the integers '''Z''' or the [[p-adic integer|''p''-adic integers]] '''Z'''_''p''.<ref>[[Localization of a ring#Terminology|away from 2]], that is, if 2 is invertible in the ring, quadratic forms are equivalent to [[symmetric bilinear form]]s (by the [[polarization identities]]), but at 2 they are different concepts; this distinction is particularly important for quadratic forms over the integers.</ref> [[Binary quadratic form]]s have been extensively studied in [[number theory]], in particular, in the theory of [[quadratic field]]s, [[continued fraction]]s, and [[modular forms]]. The theory of integral quadratic forms in ''n'' variables has important applications to [[algebraic topology]].