# Quadratic forms

Course Code : | 2000WETQUF |

Study domain: | Mathematics |

Bi-anuall course: | Taught in academic years starting in an even year |

Academic year: | 2017-2018 |

Semester: | 1st semester |

Contact hours: | 60 |

Credits: | 6 |

Study load (hours): | 168 |

Contract restrictions: | No contract restriction |

Language of instruction: | English |

Exam period: | exam in the 1st semester |

Lecturer(s) | Karim Johannes Becher Andrew Dolphin |

### 1. Prerequisites *

an active knowledge of

- Dutch
- English

general notion of the basic concepts of

Algebra as covered by the bachelor program.

### 2. Learning outcomes *

- The student masters the basic notions and the central theorems from the theory of quadratic forms over fields.
- He/she can be apply these to simple questions (e.g. concerning sums of squares) over particular fields.
- He/she has the necessary background to understand central questions of current researchand to read a research papers in this area.

### 3. Course contents *

A quadratic form is a homogeneous polynomial of degree 2. The study of quadratic forms over arbitrary fields of characteristic different from two was initiated by Ernst Witt in 1936. He showed that the properties of quadratic forms over a given field are characterised by a commutative ring, the socalled Witt ring of the field. In the 1960s Albrecht Pfister discovered the importance of stongly multiplivative forms (now called Pfister forms) in this theory. After this discovery and the fomulation of structure theorems for the Witt ring, the quadratic form theory got much more scientific attention. Since then, until Vladimir Voevodsky prooved the Milnor conjecture in 2002, it remained an open question if the graded Witt ring of any field (the graduation is induced by the powers of the fundamental ideal of the Witt ring) can be descibed in terms of generators and relations.

Quadratic form theory applies to the study of sums of squares and field orderings. Furthermore, it is related to many other areas of mathematics, in particular the study of central simple algebras and linear algebraic groups.

The chapters covered in the course are the following:

- Symmetric bilinear forms (orthogonal sum, tensor product, isotropy, Witt's Theorems, decomposition, reflections, orthogonal group, multiplicative forms, level of a field)
- Quadratic forms (correspondence with symmetric bilinear forms, complications in characteristic 2, anisotropic part, discriminant, quaternion algebras)
- Field extensions (behaviour of forms, odd degree extensions, quadratic extensions, rational function field, Cassels-Pfister Theorem, function field of a quadratic form, characterisation of Pfister forms)
- Witt rings (construction, examples, spectrum and other ring theoretic properties, field orderings and signatures, Sylvester's inertia theorem, Artin-Schreier, Pfister's Local-Global Principle)
- Invariants and Milnor's conjecture

### 4 International dimension*

The course has an international dimension.

### 5. Teaching method and planned learning activities

Personal work

### 6. Assessment method and criteria

The intermediate test and the participation cannot be repeated. In the case where the resit exam is taken, the final grade for the second session is computed in the same way (see 5.3), on the basis of the original grades for participation and the test.

Examination

Continuous assessment

### 7. Study material *

#### 7.1 Required reading

Course notes and exercise sheets

**7.2 Optional reading**

The following study material can be studied voluntarily :### 8. Contact information *

Karim Johannes Becher (office G.224); email: KarimJohannes.Becher@uantwerpen.be

Andrew Dolphin (office G.320a); email: Andrew.Dolphin@uantwerpen.be