Editing Quadratic form (section)

=== Generalization ===
Let {{math|''R''}} be a [[commutative ring]], {{math|''M''}} be an {{math|''R''}}-[[Module (mathematics)|module]], and {{math|''b'' : ''M'' × ''M'' → ''R''}} be an {{math|''R''}}-bilinear form.{{refn|The bilinear form to which a quadratic form is associated is not restricted to being symmetric, which is of significance when 2 is not a unit in {{math|''R''}}.}} A mapping {{math|''q'' : ''M'' → ''R'' : ''v'' ↦ ''b''(''v'', ''v'')}} is the ''associated quadratic form'' of {{math|''b''}}, and {{math|''B'' : ''M'' × ''M'' → ''R'' : (''u'', ''v'') ↦ ''q''(''u'' + ''v'') − ''q''(''u'') − ''q''(''v'')}} is the ''polar form'' of {{math|''q''}}.

A quadratic form {{math|''q'' : ''M'' → ''R''}} may be characterized in the following equivalent ways:
* There exists an {{math|''R''}}-bilinear form {{math|''b'' : ''M'' × ''M'' → ''R''}} such that {{math|''q''(''v'')}} is the associated quadratic form.
* {{math|1=''q''(''av'') = ''a''<sup>2</sup>''q''(''v'')}} for all {{math|''a'' ∈ ''R''}} and {{math|''v'' ∈ ''M''}}, and the polar form of {{math|''q''}} is {{math|''R''}}-bilinear.