Enrollment year
2021/2022
Academic discipline
MAT/02 (ALGEBRA)
Department
DEPARTMENT OF MATHEMATICS "FELICE CASORATI"
Curriculum
PERCORSO COMUNE
Period
2nd semester (01/03/2023 - 09/06/2023)
Lesson hours
60 lesson hours
Activity type
WRITTEN AND ORAL TEST
Prerequisites
The courses of Linear algebra and Algebra 1.
Learning outcomes
The course is an introduction to Galois theory, with some complements of group theory and of the theory of modules over a ring.
Course contents
Modules over a ring.
Finitely generated abelian groups.
Group actions on sets; group representations.
Sylow theorems.
Field extensions; algebraic and transcendental elements. Ruler and compass constructions. 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.
Teaching methods
Lectures and exercise sessions
Reccomended or required readings
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.
Assessment methods
The exam consists of a written test, during which the student must solve some exercises, and of an oral examination, during which the student must answer some questions, mainly of a theoretical nature.
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