Integral quadratic form

Builds on Quadratic form Integers Matrix

Definition

Let $n \ge 1$ be an integer. An integral quadratic form in $n$ variables is a quadratic form $Q(x_1, \dots, x_n) = \sum_{1 \le i \le j \le n} a_{ij} x_i x_j$ whose coefficients $a_{ij}$ all lie in the integers. Two integral quadratic forms $Q$ and $Q'$ in $n$ variables are equivalent if $Q'(x) = Q(Mx)$ for some $n \times n$ integer matrix $M$ with $\det M = \pm 1$ .

Remarks

Every integral quadratic form takes integer values at integer points. The form can be written $Q(x) = \frac12 x^T A x$ with a symmetric integer matrix $A$ whose diagonal entries $2a_{ii}$ are even. Some authors call $Q$ integral only when $Q(x) = x^T B x$ for a symmetric integer matrix $B$ . This means that all cross coefficients $a_{ij}$ with $i < j$ are even; such forms are called classically integral.

Example: $x^2 + xy + y^2$ is integral, with $a_{11} = a_{12} = a_{22} = 1$ ; it is not classically integral, because the coefficient of $xy$ is odd.

Non-example: $x^2 + \frac12 xy + y^2$ is a quadratic form over $\Q$ but not integral, since the coefficient of $xy$ is not an integer; its value at $(1, 1)$ is $5/2$ .

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Integers object
Matrix concept
Quadratic form concept
Integral quadratic form definition

Tag	`TA5025GWDQ`
URI	`m2://TA5025GWDQ`
Kind	definition
Writing	W0 stub
MSC	11E12, 11E20
Defines	Integral quadratic form
Uses	Quadratic form, Integers, Matrix
Changed	2 November 2026

Integral quadratic form

Definition

Remarks

Reading path

Further reading

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