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Fundamental Algebra

Code: M501     Acronym: M501

Keywords
Classification Keyword
OFICIAL Mathematics

Instance: 2021/2022 - 1S Ícone do Moodle

Active? Yes
Responsible unit: Department of Mathematics
Course/CS Responsible: Doctoral Program in Mathematics

Cycles of Study/Courses

Acronym No. of Students Study Plan Curricular Years Credits UCN Credits ECTS Contact hours Total Time
IUD-M 4 PE do Prog Inter-Univ Dout Mat 1 - 9 60 243

Teaching language

English

Objectives

Introduction to basic topics of abstract algebra.

Learning outcomes and competences

Familiarity with basic concepts and results of Abstract Algebra.

Working method

Presencial

Program

1) Groups: permutations, Lagrange theorem, homomorphisms, quotient group, group actions.

     Finite Abelian groups (Direct sums and the fundamental theorem), the Sylow Theorems, the Jordan Hoçder theorem. Presentations and the Nielson-Schreier theorem.

2) Commutative rings and fields: Polynomials, homomorphisms, quotient rings and finite fields. Fundamental theorem of Galois Theory.

     Prime and maximal ideals. Unique factorization domains, Noetherian rings, primary decomposition and the lasker Noether Theorem.

3) Rings and modules: Free modules, projective and injective modules.

     Chain conditions and semisimple rings.

Depending on the background and interests of the students, some topics may be considerably more developed than others.

Mandatory literature

Rotman, J.J.; Advanced Modern algebra, ams, 2010. ISBN: 978-0-8218-4741-1
Pierre Antoine Grillet; Abstract Algebra, springer, 2007. ISBN: 978-0387715674

Complementary Bibliography

Nathan Jacobson; Basic algebra. ISBN: 0-7167-0453-6 (Vol. I)
Nathan Jacobson; Basic algebra II, 2009. ISBN: 978-0486471877
Thomas W. Hungerford; Algebra. ISBN: 0-387-90518-9
Serge Lang; Algebra, springer, 2002. ISBN: 978-1-4612-6551-1
I. Martin Isaacs; Algebra. ISBN: 0-534-19002-2
I. N. Herstein; Topics in ring theory. ISBN: 0-226-32802-3
Hideyuki Matsumura; Commutative ring theory. ISBN: 0-521-25916-9

Teaching methods and learning activities

The course material is presented and developed in the lectures.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation Weight (%)
Exame 40,00
Trabalho escrito 60,00
Total: 100,00

Amount of time allocated to each course unit

designation Time (hours)
Frequência das aulas 60,00
Total: 60,00

Eligibility for exams

Course registration is the only requirement.

Calculation formula of final grade

There will be 2 written works, T1 and T2 classified from 0 to 20. If E is the score of the final exam (from 0 to 20), the final score will be given by the formula

0.3*(T1+T2)+0.4*E

The students may be asked during the classes to explain some of the solutions they submitted.

Special assessment (TE, DA, ...)

Exam that will count for a 100% of the grade.

Classification improvement

Exam that will count for a 100% of the grade.
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