Algebra - ELEMENTI DI ALGEBRA (2020/2021)

Course code
4S00022
Name of lecturers
Fabiano Bonometti, Enrico Gregorio
Number of ECTS credits allocated
6
Academic sector
MAT/02 - ALGEBRA
Language of instruction
Italian
Location
VERONA
Period
II semestre dal Mar 1, 2021 al Jun 11, 2021.

To show the organization of the course that includes this module, follow this link * Course organization

Lesson timetable

Go to lesson schedule

Learning outcomes

The course provides an introduction to modern algebra. After presenting and discussing the main algebraic structures (groups, rings, fields), the focus is on Galois theory. Also some applications are discussed, in particular results on solvability of polynomial equations by radicals. At the end of the course the student will be expected to demonstrate that s/he has attained adequate skills in synthesis and abstraction, as well as the ability to recognize and produce rigorous proofs and to formalize and solve moderately difficult problems related to the topics of the course.

Syllabus

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MM: Elementi di algebra teoria
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Groups, subgroups, cosets, quotient groups. Cyclic groups. The symmetric group. Sylow's Theorems. Solvable groups. Rings. Ideals. Homomorphisms. Principal ideal domains. Unique factorization domains. Euclidean rings. The ring of polynomials. Fields. Algebraic field extensions. The splitting field of a polynomial. Finite fields. Constructions with ruler and compass.
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MM: Drills on elements of algebra
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MM: Galois theory
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Normal extensions. Separable extensions. Galois theory. Theorem of Abel-Ruffini.
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MM: Drills on Galois theory
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Reference books
Author Title Publisher Year ISBN Note
I. N. Herstein Algebra Editori Riuniti 2003
Sigfried Bosch Algebraic Geometry and Commutative Algebra Springer 2013

Assessment methods and criteria

The exam consists of a written examination. The mark obtained in the written examination can be improved by the mark obtained for the homework and/or by an optional oral examination. Only students who have passed the written exam will be admitted to the oral examination.