Knowledge and understanding: acquiring the basic notions of group theory, with particular regard to finite group theory; understanding the properties of field extensions and Galois theory; knowing the main applications of Galois theory. Applying knowledge and understanding: carrying out exercises about group theory and Galois theory; analyzing the properties of a group which has an explicit representation or a small order; computing the Galois groups of a field extension; analyzing a field extension by using Galois theory. Making judgements: being able to decide whether a given problem can be dealt with the tools introduced in the course. Communication skills: presenting in a correct and appropriate manner definitions and theorems of group theory and of Galois theory Learning skills: reading and understanding texts concerning group theory and Galois theory.

Elementary knowledge about groups, fields and field extensions.

Group theory: basic notions of group theory, actions of groups on sets or groups, permutation groups, Cayley's theorem, p-groups, Sylow theorems, semidirect product, conjugacy relation, finitely generated abelian groups, solvable groups, derived series, and composition series. Galois theory: splitting fields, derivative of a polynomial, perfect fields, normal and separable extensions, Galois group, fundamental theorem of Galois theory, ruler and compass constructions, symmetric rational functions, Galois criterion for the solvability of polynomial equations by radicals, Abel-Ruffini theorem.

M. Suzuki "Group Theory I" - Springer
I.H. Herstein "Algebra" - Editori Riuniti
N. Jacobson "Basic Algebra I" - W.H.Freeman and Company
I. Stewart, "Galois Theory" - Chapman & Hall

Group theory. Basic notions of group theory. Homomorphisms, subgroups, generators of a subgroup, centralizers and normalizers, order of a group and of an element, cosets of a subgroup, normal subgroups and quotient groups, first and second homomorphism theorem and consequences. Group actions on sets or groups. Orbits and stabilizers. Equations of classes. Permutation groups. Cayley's theorem. Conjugacy relation in a group. Definition and properties of p-Groups. Sylow theorems and consequences. (Inner and outer) Semidirect product. Finite abelian groups and their rank. Subgroups of free abelian groups. Fundamental theorem of finitely generated abelian groups. Solvable groups. Derived series. Composition series. Jordan-Holder theorem. Galois theory. Splitting fields. Perfect fields. Derivative of a polynomial. Normal and separable extensions. Galois group. Fundamental theorem of Galois theory. Ruler and compass constructions. Symmetric rational functions. Galois criterion for the solvability of polynomial equations by radicals. Characterization of irreducible polynomials via the Galois group.Abel-Ruffini theorem.

During the lesson, theoretical aspects and exercises are presented using the blackboard. Students are strongly invited to actively take part in the lessons. Regularly we assign to the students some exercises as homework.

Teacher's notes, homework assignments, and some texts of previous written examinations are available on the MOODLE platform.

The written test consists in solving exercises modeled on those solved during the lectures or given as homework. A score not less than 16/30 gives access to the oral test which has to be passed in the same exam session in which the written test is passed. Handing in any written test substitutes a possible preceding one. During the oral test, the student has to be able to explain the constructions and the definitions introduced during the course and present the proof of the theorems explained during the lessons. The final score depends on both written and oral tests. To ensure the access to aids at the exam from students with disabilities, specific learning disorder (SLD), or special educational needs (SED), please preemptively contact the University's Disability Service or SLD Service.