THe student should know how to recognize a group as a semidirect product, to classify groups of appropriate orders, to distinguish normal field extension, separable ones, to compute the degree of a finite extension, to determine the Galois group of an extension in reasonably simple cases.
Course Prerequisites
The courses of Linear algebra and Algebra 1.
Teaching Methods
Lectures and exercise sessions
Assessment Methods
The exam consists of a written and an oral part. The written part aims at evaluating the abilities that the student has acquired in the computations and in the resolution of problems connected with the subject of the class. The exercises will include questions of varying difficulty aiming at determining the degree of knowledge and competence the student has arrived at. To pass to the oral part it is necessary to get a mark of at least 15/30 in the written part. The goal of the oral part will be to determine the degree to which the student has learnt the notions presented in the class, the clarity with which he/she is able to explain them and the skill with which he/she is able to apply them. The final mark will be determined from the overall extent and the depth of the learning, the clarity of exposition and the skill in the solution to the problems. The mark will not reduce to an arithmetic mean, but will emerge from a comparison between the written and the oral part.
Texts
I.N. Herstein, "Algebra", Editori Riuniti. J.S. Milne, "Group Theory", http://www.jmilne.org/math/CourseNotes/gt.html. D.J.H. Garling, "A Course in Galois Theory", Cambridge University Press. J.S. Milne, "Fields and Galois Theory", http://www.jmilne.org/math/CourseNotes/ft.html. M. Artin, "Algebra", Bollati Boringhieri. P. Aluffi, "Algebra: chapter 0", American Mathematical Society.
Contents
The course consists of two parts: the first contains some results from group theory, the second is an introduction to Galois theory. First part: finitely generated abelian groups, group actions on sets; semidirect products, Sylow theorems. Second part: field extensions; algebraic and transcendental elements, splitting fields of polynomials, algebraic closure of a field, normal, separable and Galois extensions, fixed fields and Galois groups; the fundamental theorem of Galois theory, polynomials solvable by radicals.
Course Language
Italian
More information
The students belonging to the categories of the project on innovative teaching will have the possibility to attend office hours also in the late afternoon and to see the notes of the lectures.
