Intro to Quadratic Forms

The theory of symmetric bilinear forms is intrinsically linked to the theory of quadratic forms. Over rings where two is a unit, they are even the same.

Definition I.4.1

A quadratic form on a -module is a function such, that , for all , and such, that the function on defined by is a bilinear form over . The pair is then called a quadratic module.

Definition I.4.2

Just as for bilinear modules, an isometry between two quadratic modules is an homomorphism between the two modules which retains the structure of the quadratic modules. Two quadratic modules are called isometric if there exists a bijective isometry between them.

Remark

Just as for bilinear forms, isometries are also isomorphisms in the category of quadratic modules and we therefor also say that two quadratic modules are isomorphic instead of isometric.

Remark

An isometry of quadratic modules is also an isometry of the underlying bilinear modules, but the converse only holds if is not a zero divisor in .

Remark

If , then the following maps where , are isomorphisms, inverse to each other, between the -modules of quadratic and bilinear forms on a given module.

It is easy to see that if is a not necessarily symmetric bilinear form, then the function is quadratic with .

Theorem I.4.3

If is finitely generated and projective, every quadratic form is of the form described above.

Remark

Note that two bilinear forms give rise to the same quadratic form if, and only if, their difference is symplectic.

As for bilinear modules, we can construct the orthogonal sum of quadratic modules:

Definition I.4.4

For quadratic modules the orthogonal sum is defined as the direct sum of the modules with a quadratic form defined by

Definition I.4.5

We call a quadratic space if is finitely generated and projective, and if, in addition, the underlying bilinear form is regular.

Similar to for a Gram-Matrix of a bilinear module, we will use the notation to define isometry classes of quadratic modules. As such, we write for the quadratic module with quadratic form . The bilinear module associated with this is .

We now want to investigate what information the determinant of a quadratic module holds. Using basic linear algebra, particularly the Leibniz-rule, we get the following:

Lemma

The determinant of a matrix , with n odd, is of the form: , where with unknowns.

Definition and Remark I.4.6

Let be a finitely generated and free quadratic module with odd dimension and bases and , with , then:

Similar to bilinear modules, we have the following

Definition I.4.7

Let be a finitely generated free quadratic module of odd dimension n. The half-determinant of is the class modulo squares of units of the half-discriminant of any basis of , written as The quadratic module is called half-regular if its half-determinant is represented by a unit in .

Remark

Over a field of characteristic two, a quadratic space of odd dimensions cannot be regular, as the underlying bilinear form must then be symplectic.

We now give a first decomposition theorem over fields:

Theorem 1.4.8

Let be a regular quadratic space over a field of characteristic not equal two, then can be decomposed into orthogonal subspaces as follows: , where the are regular spaces of dimension two, the are one-dimensional spaces with and .

Literature

Symmetric Bilinear Forms by J. Milnor and D. Husemoller
Arithmetic Theory of Quadratic Forms by R. Scharlau