Enrollment year
2017/2018
Academic discipline
MAT/02 (ALGEBRA)
Department
DEPARTMENT OF MATHEMATICS "FELICE CASORATI"
Curriculum
PERCORSO COMUNE
Period
2nd semester (04/03/2019 - 14/06/2019)
Lesson hours
56 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; submodules and quotient modules. Module homomorphisms; homomorphism and isomorphism theorems for modules. The structure theorem for finitely generated modules over a principal ideal domain.
Finitely generated abelian groups. Group actions on sets; group representations. The class equation. Cauchy theorem and Sylow theorems. Semidirect products of groups. Soluble groups.
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. Galois theory for finite fields. The theorem of the primitive element. Cubics and quartics; polynomials solvable by radicals.
Teaching methods
Lectures and exercise sessions
Reccomended or required readings
I.N. Herstein: "Algebra", Editori Riuniti.
M. Artin: "Algebra", Bollati Boringhieri.
P. Aluffi: "Algebra: chapter 0", American Mathematical Society.
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.
I.N. Stewart: "Galois Theory", CRC Press.
J.S. Milne: "Fields and Galois Theory", http://www.jmilne.org/math/CourseNotes/ft.html.
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.
Further information
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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