# Algebra (2019/2020)

Course code
4S00022
Credits
9
Coordinator
Lidia Angeleri
Other available courses
Other available courses
MAT/02 - ALGEBRA
Language of instruction
Italian
Teaching is organised as follows:
Activity Credits Period Academic staff Timetable
Elementi di algebra teoria 5 II semestre Lidia Angeleri

Elementi di algebra esercitazioni 1 II semestre Fabiano Bonometti

Teoria di Galois teoria 2 II semestre Lidia Angeleri

Teoria di Galois esercitazioni 1 II semestre Fabiano Bonometti

### 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: Elementi di algebra esercitazioni
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MM: Teoria di Galois teoria
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Normal extensions. Separable extensions. Galois theory. Theorem of Abel-Ruffini.
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MM: Teoria di Galois esercitazioni
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### 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.

 Activity Author Title Publisher Year ISBN Note Elementi di algebra teoria I. N. Herstein Algebra Editori Riuniti 2003 Elementi di algebra teoria Sigfried Bosch Algebraic Geometry and Commutative Algebra Springer 2013