ALGEBRA
Academic year and teacher
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- Versione italiana
- Academic year
- 2022/2023
- Teacher
- CLAUDIA MENINI
- Credits
- 15
- Didactic period
- Annualità Singola
- SSD
- MAT/02
Training objectives
- The aim of the course is to provide some basic concepts in Algebra such as sets, maps, quotient sets, groups, group homomorphisms, quotient groups, rings, homomorphisms, quotient rings, ring of polynomial, factorial domains, principal ideal domains, field of quotients of a domain, field extensions.
At the end of the course the student is supposed to have achieved logical skills, appropriate scientific language
and a correct mathematical formalism which will allow him/her to solve basic problems concerning fundamental algebraic structures such as cyclic groups, quotient groups, product of groups, quotient rings, simple algebraic extensions.
At the end, the student will also be able to attend courses about Rings and Modules, Categories and Galois Theory. Prerequisites
- None
Course programme
- General results on sets. Cardinality. Peano's axioms and natural numbers. Binary operations. Grupoids. Semigroups. Groups. Group homomorphisms, Kernel and Image. Correspondence Theorem for groups. The Symmetric Group and the Diehedrial Group. Direct product of groups. (34 hours)
Equivalence relations and Partitions. Quotient Sets. Congruences and Lagrange's Theorem. Normal subgroups and Quotient Groups. Fundamental Theorem for Quotient Groups and Isomorphism Theorems. Cyclic Groups. Representations of a Group. Sylow's Theorems and Applications. (20 hours)
Rings and ring homomorphisms. Ideals. Correspondence Theorem for Rings. Fundamental Theorem for Quotient Rings and Isomorphism Theorems. Characteristic of a ring. Fermat's Theorem. The ring of integers. Direct Product of Rings. Chinese Remainder Theorem. Systems of congruences. (20 hours).
Integral domains, fields. Prime and maximal ideals. Zorn's Lemma. Krull's Lemma. Axiom of Choice. Ring of formal power series, ring of polynomials and its Universal Property. Division of polynomials. Ruffini's Theorem. Polynomial functions. Ideals generated by subsets. Euclidean domains and Principal Ideals Domains. Ring of Gaussian integers. Factorial Domains. The Quotient Field of a Domain. The field of rational numbers. The ring of polynomials with coefficients in a factorial domain is a factorial domain. Eisenstein's criterion. (22 hours)
Field extensions, simple extensions. Characterization of simple algebraic extensions and of finite extensions. Algebraically closed fields and Fundamental Theorem of Algebra (statement). Finite fields. (24 hours) Didactic methods
- All concepts which appear in the program of the course will be introduced and explained during lessons of the course. Also all Theorems, and their prerequisites, which appear in the program of the course will be proved with all needed details. Examples will be given to illustrate a new concept and/or a new result. To stimulate student’s interest, a number of exercises will be given during the lectures. Willing student can give their solution to the teacher for eventual correction and discussion.
Learning assessment procedures
- The traditional exam consists of a written examination and an oral examination. These examinations are aimed at verifying the learning themes of the course. More in detail the written exam is designed to examine the ability of solving exercises standard coursework, while the oral exam aims to check the understanding even of the theoretical part of the course, with particular reference to the logical sequence of results.
As an alternative, in-course partial exams are provided accordingly to the following scheme. 2 written partial exams: one at the end of the first semester, in the period of interruption of the lessons,and one at the end of the course; 2 oral partial exams: one in the period of interruption of classes in February, that focuses on the first part of the program, and one at the end of the course that focuses on the second part of the course. The last oral examination can be carried out at the request of the parties concerned, in a subsequent period.
The written exam consist of five exercises that with a mark ranging from 0 up to seven.
The student can chose to take on only written partial exams exempting him/her from the traditional written examination. Reference texts
- C. Menini and F. Van Oystaeyen, "Abstract Algebra, A comprehensive Treatment", Marcel Dekker.
T.W. Hungerford - Algebra - Springer-Verlag
P.M. Cohn - Algebra Vol. I - J. Wiley & Sons
N. Jacobson - Basic Algebra I - Freeman
M. Artin - Algebra - Prentice Hall
