# Fields and Galois theory

Course Code : | 1002WETKLG |

Study domain: | Mathematics |

Academic year: | 2017-2018 |

Semester: | 1st semester |

Sequentiality: | - |

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) | Boris Shoykhet |

### 1. Prerequisites *

At the start of this course the student should have acquired the following competences:

an active knowledge of

general notion of the basic concepts of

specific prerequisites for this course

an active knowledge of

- English

- general knowledge of the use of a PC and the Internet

general notion of the basic concepts of

The student knows the basic concepts of set theory and of linear algebra.

specific prerequisites for this course

Content of the course `Groepen en Ringen' ('Groups and Rings'). The student should be familiar with concepts of vector spaces, fields, groups. The student has some basic knowledge on rings.

### 2. Learning outcomes *

- The student knows the most fundamental concepts and theorems from field theory and classical Galois theory.
- The student is familiar with diverse properties of field extensions. The student can compute the Galois group for several classical situations.
- The student understands the connection between the concept of field extensions and Galois theory, on the one hand, and the problems of solving polynomial equations and of ruler and compass constructions, on the other hand.

### 3. Course contents *

- Rings of polynomials, the procedure of joining a root of a polynomial to the field.
- Algebraic field extensions, normal extensions.
- Definition and construction of an algebraic closure of a field.
- Basis of Galois theory, Galois correspondence between subextensions and subgroups.
- Cyclotomic field extensions.
- Applications of Galois theory: solving of concrete polynomial equations, constructions with ruler and compass.
- Proof of the Fundamental Theorem of Algebra.

### 4 International dimension*

The course has an international dimension.

### 5. Teaching method and planned learning activities

Class contact teachingLectures Practice sessions

Personal workExercises Assignments Individually

Personal work

### 6. Assessment method and criteria

ExaminationWritten with oral presentation Closed book Open-question

Continuous assessmentExercises (Interim) tests Participation in classroom activities

Continuous assessment

### 7. Study material *

#### 7.1 Required reading

Course notes will be provided at the Blackboard.

**7.2 Optional reading**

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

Boris Shoikhet (G.115); email: boris.shoikhet@uantwerpen.be

Stijn Symens (lokaal G.308); email: stijn.symens@uantwerpen.be