# 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

### 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

### Remark

If

It is easy to see that if

### Theorem I.4.3

If

### 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

### Definition I.4.5

We call

Similar to

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

### Definition and Remark I.4.6

Let

, , .

Similar to bilinear modules, we have the following

### Definition I.4.7

Let

### 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

## Literature

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