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Fundamental Algebra
| Code: | M501 | Acronym: | M501 |
| Keywords | |
|---|---|
| Classification | Keyword |
| OFICIAL | Mathematics |
Instance: 2023/2024 - 1S 
| 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 |
Fields changed: Mandatory literature, Componentes de Avaliação e Ocupação
Teaching language
EnglishObjectives
Introduction to fundamental topics in abstract algebra.Learning outcomes and competences
Familiarity with basic concepts and results of abstract algebra (groups, commutative rings and fields).Working method
PresencialProgram
The program focuses on the theory of not necessarily commutative rings and modules, with applications to group theory, (multi)linear algebra and geometry.
1. Rings and modules, chain conditions, composition series and the Hilbert Basis Theorem.
2. Applications of set theory and Zorn's lemma: existence of bases of vector spaces, Hamel bases; sums and products of families, free modules; semisimplicity, Maschke's Theorem; existence of an algebraic closure of a field; bases and degree of transcendence of an extension of fields.
3. Finitely generated modules over PIDs and applications to linear algebra: canonical forms and Smith's normal form; Law of Inertia; orthogonal and simplicial groups, unitary groups.
4. Categories of modules and functors; limits; projective and injective modules; divisible abelian groups; tensor products; flat modules; Dedekind rings.
5. Commutative algebra and algebraic geometry: affine and irreducible varieties, radical ideals, Nullstellensatz, Grobner bases.
Other topics may be covered, depending on the availability of time and the interest of the students.
1. Rings and modules, chain conditions, composition series and the Hilbert Basis Theorem.
2. Applications of set theory and Zorn's lemma: existence of bases of vector spaces, Hamel bases; sums and products of families, free modules; semisimplicity, Maschke's Theorem; existence of an algebraic closure of a field; bases and degree of transcendence of an extension of fields.
3. Finitely generated modules over PIDs and applications to linear algebra: canonical forms and Smith's normal form; Law of Inertia; orthogonal and simplicial groups, unitary groups.
4. Categories of modules and functors; limits; projective and injective modules; divisible abelian groups; tensor products; flat modules; Dedekind rings.
5. Commutative algebra and algebraic geometry: affine and irreducible varieties, radical ideals, Nullstellensatz, Grobner bases.
Other topics may be covered, depending on the availability of time and the interest of the students.
Mandatory literature
Joseph J. Rotman; Advanced modern algebra. ISBN: 978-1-4704-1554-9 : 1a pt.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 examAssessment Components
| designation | Weight (%) |
|---|---|
| Apresentação/discussão de um trabalho científico | 40,00 |
| Trabalho prático ou de projeto | 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
Individual homework problems will be assigned throughout the semester and these must be submitted for assessment. This accounts for 60% of the final grade.Each student has to prepare, in written form, a small monograph on a scientific topic agreed on with the lecturer, and present it orally. This accounts for 40% of the final grade.
Special assessment (TE, DA, ...)
Exam for 100% of the grade.Classification improvement
Exam for 100% of the grade.
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