Editing Quadratic form (section)

=== Historical use ===
Historically there was some confusion and controversy over whether the notion of '''integral quadratic form''' should mean:
; ''twos in'' : the quadratic form associated to a symmetric matrix with integer coefficients
; ''twos out'' : a polynomial with integer coefficients (so the associated symmetric matrix may have half-integer coefficients off the diagonal)
This debate was due to the confusion of quadratic forms (represented by polynomials) and symmetric bilinear forms (represented by matrices), and "twos out" is now the accepted convention; "twos in" is instead the theory of integral symmetric bilinear forms (integral symmetric matrices).

In "twos in", binary quadratic forms are of the form {{math|''ax''<sup>2</sup> + 2''bxy'' + ''cy''<sup>2</sup>}}, represented by the symmetric matrix
<math display="block">\begin{pmatrix}a & b\\ b&c\end{pmatrix}</math>
This is the convention [[Gauss]] uses in ''[[Disquisitiones Arithmeticae]]''.

In "twos out", binary quadratic forms are of the form {{math|''ax''<sup>2</sup> + ''bxy'' + ''cy''<sup>2</sup>}}, represented by the symmetric matrix
<math display="block">\begin{pmatrix}a & b/2\\ b/2&c\end{pmatrix}.</math>

Several points of view mean that ''twos out'' has been adopted as the standard convention. Those include:
* better understanding of the 2-adic theory of quadratic forms, the 'local' source of the difficulty;
* the [[lattice (group)|lattice]] point of view, which was generally adopted by the experts in the arithmetic of quadratic forms during the 1950s;
* the actual needs for integral quadratic form theory in [[topology]] for [[intersection theory]];
* the [[Lie group]] and [[algebraic group]] aspects.