### General Information

Course Code | XM_0058 |
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Credits | 6 EC |

Period | P1+2 |

Course Level | 400 |

Language of Tuition | English |

Faculty | Faculty of Science |

Course Coordinator | prof. dr. R.M.H. de Jeu |

Examiner | dr. S.R. Dahmen |

Teaching Staff |
prof. dr. R.M.H. de Jeu dr. S.R. Dahmen |

### Practical Information

You need to register for this course yourself

Last-minute registration is available for this course.

Teaching Methods | Lecture |
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Target audiences

This course is also available as:

### Course Objective

* The student knows an advanced topic in algebra, in this case part ofthe theory of Brauer groups. This will include most of the following:

+ the theory of central simple (finite dimensional) algebras over a

field, including the Brauer group;

+ Galois cohomology, and its relation to the Brauer group;

+ cyclic division algebras;

+ norms on algebras;

+ division algebras over local fields;

+ division algebras over number fields.

* The student can independently learn part of this material, and give a

talk about it.

* The student can independently set an assignment on that part, provide

model solutions, and mark the work handed in.

### Course Content

Over the real numbers, the only finite dimensional division algebras arethe real numbers, the complex numbers, and the Hamiltonians. Over any

field k, the problem can be studied by means of the Brauer group of k.

For this one considers finite dimensional k-algebras with only the

obvious two ideals, and as centre k. Any such algebra is isomorphic to a

matrix ring over a division ring over k. Under the tensor product over

k, such algebras do not form a group as there is no inverse. But on a

set of equivalence classes defined using matrix sizes, one can put a

group structure, which gives the Brauer group. By studying the structure

of the algebras involved, this group can be linked with group cohomology

of finite Galois extensions of k (Galois cohomology). This makes the

group more computable for certain types of fields, in particular for

finite extensions of the p-adic numbers or the rationals.

The non-commutative case has several surprising properties. For example,

any automorphism of the type of algebra considered is induced by

conjugation with a unit of the algebra. The techniques involved are

therefore also often quite different from, e.g., the ones used in field

theory or Galois theory.

Brauer groups found an application in diophantine geometry, where they

can sometimes be used to show a variety has no rational point over a

given number field (the Brauer-Manin obstruction).

### Teaching Methods

A seminar, for 2 hours per week during each week of periods 1 and 2.Each student gives one or two talks. The student also creates a

corresponding assignment for the other students (which has to be

approved by the coordinators), and grades the corresponding solutions of

the other students based on (approved) etailed model solutions. This

grading is checked at random by the coordinators.

### Method of Assessment

The grade is determined as follows:* 30% by the talk(s) of the student;

* 30% by the understanding of the material by the student (both for the

talk and the preparation of the assignment and the model solution);

* 40% by the grades of the work handed in.

There will be a minimum attendance requirement in order to get a grade

for this course. The exact requirement will be determined at the

beginning of the seminar.

Due to the nature of the seminar, there will not be a resit.

### Literature

We shall work mostly from `Associative algebras', Graduate texts inmathematics volume 88, Springer Verlag. (There is a 2012 reprint of the

original 1982 edition.) It contains most of the material we want to

treat, e.g., central simple algebras, Brauer groups, group cohomology,

relation between Brauer groups and Galois cohomology, completions of

finite dimensional algebras, the Brauer groups of local fields and of

number fields, which corresponds to Chapters 12 through 18. Several of

those topics are covered in other texts as well, and we may use one or

more of those during the seminar.

### Target Audience

mMath### Additional Information

The seminar will only run if there are at least ten participatingstudents.

### Recommended background knowledge

The VU courses Linear Aagebra, Group theory, and Rings and fields, aswell as basic knowledge of module theory (as in the intensive course of

mastermath), or equivalent. Specifically, the relevant material (and

much more) can be found in Chapters 1 through 4, 7 through 11, and

Chapter 13, of Dummit and Foote, Abstract Algebra, third edition, but

also in many other places.

At some stage the course Galois theory will be necessary as well (the

material contained in Section 14.1 and 14.2 of the book by Dummit and

Foote).Knowledge of p-adic numbers can help provide background for

Brauer groups of p-adic fields, but as completion in the non-commutative

case has its own peculiarities, we have to treat it anyway. Algebraic

number theory can be useful but is not required.