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TOPICS IN ADVANCED ALGEBRA

Degree course: 
Corso di Second cycle degree in MATHEMATICS
Academic year when starting the degree: 
2023/2024
Year: 
1
Academic year in which the course will be held: 
2023/2024
Course type: 
Compulsory subjects, characteristic of the class
Credits: 
8
Period: 
Second semester
Standard lectures hours: 
64
Detail of lecture’s hours: 
Lesson (64 hours)
Requirements: 

Knowledge of basic algebraic structures and their properties: groups, rings, polynomials, fields. Knowledge of basic results in linear algebra and matrix calculus.

Final Examination: 
Orale

Written and oral exam.
The written examination lasts 2 hours and 30 minutes and tipically consists of 2 or 3 exercises divided in subquestions.
The oral examination starts immediately after the written part and usually begins with the discussion of the written test. The student will then be required to present some of the results seen in class. It will be object of evaluation the ability to present a proof in a complete and rigorous.

Passing examination and the final grading depend both on oral and written tests.

Assessment: 
Voto Finale

TEACHING OBJECTIVES
Knowledge of Galois Theory with applications.

EXPECTED LEARNING OUTCOMES

At the end of the course the student will be able to:
- compute Galois group of a given polynomial or field extension
- obtain knowledge about a polynomial or a field extension from its Galois group

Ruler and compass constructions. Splitting field of a polynomial. Multiple roots. Perfect fields. (20 hours)

The Galois group. The Galois correspondence. Normal and separable extensions of a field. (20 hours)

Finite soluble groups. Simplicity of the alternating group. The criterion for the solubility by radicals of an equation. The Galois group as the permutation group of the roots of a polynomial. General equation of degree n. (20 hours)

Finite fields. (4 hours)

Ruler and compass constructions. Splitting field of a polynomial. Multiple roots. Perfect fields. (20 hours)

The Galois group. The Galois correspondence. Normal and separable extensions of a field. (20 hours)

Finite soluble groups. Simplicity of the alternating group. The criterion for the solubility by radicals of an equation. The Galois group as the permutation group of the roots of a polynomial. General equation of degree n. (20 hours)

Finite fields. (4 hours)

Frontal lectures. Attending lectures is not mandatory, but strongly recommended.
Lectures are given at the board. Every topic is explained together with exercises useful to understand and apply exposed results. Sometimes the solution is given immediately, sometimes in a subsequent lecture in order to stimulate student to autonomous work. Especially at the end of the course, lectures of summarizing exercises are scheduled, in order to choose the more appropriate method to solve an exercise and to establish connections between different topics.
The exercises presented are often taken from past written examinations: they can be found, together with other selected exercise, on the web page of the course.

For simple and short questions, ask the teacher immediately before or after the class. Email teacher for longer questions.
For further detail go to the web page of the course.

Professors

MONTI VALERIO

Borrowers