2024
Algebra I
Name: Algebra I
Code: MAT14239L
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese
Regime de Frequência: Presencial
Presentation
In this curricular unit, knowledge is acquired in different algebraic structures.
Sustainable Development Goals
Learning Goals
O1 - Knowledge: experiencing diverse algebraic structures.
O2 - Skills and Competencies: developing abstract reasoning, proof making, and the capacity of finding strategies to solve new problems.
O2 - Skills and Competencies: developing abstract reasoning, proof making, and the capacity of finding strategies to solve new problems.
Contents
Groups
Definition. Subgroups.
Cosets. Lagrange's theorem.
Normal subgroups. Quotient groups.
Group homomorphisms. Cayley's theorem.
Direct products.
Abelian finite groups.
Permutation groups.
Matrices groups.
Rings
Definition.
Subrings, ideals and quotient rings.
Ring homomorphisms.
Ring extensions.
Polynomials.
Matrices rings.
Principal domains. Euclidean domains.
Unique factorization domains.
Definition. Subgroups.
Cosets. Lagrange's theorem.
Normal subgroups. Quotient groups.
Group homomorphisms. Cayley's theorem.
Direct products.
Abelian finite groups.
Permutation groups.
Matrices groups.
Rings
Definition.
Subrings, ideals and quotient rings.
Ring homomorphisms.
Ring extensions.
Polynomials.
Matrices rings.
Principal domains. Euclidean domains.
Unique factorization domains.
Teaching Methods
Problem-solving sessions, where students are invited to work on their own or in small groups, with some moments of exposition or discussion involving the whole class.
The evaluation may be either continuous, done through between two and six partial tests and quizzes, done preferably during the classes, weighting 100% of the classification, the number of which is to be defined by the professor who is responsible for the course unit in each academic year, or by a final exam. Students who achieve a grade of 18 or above may have to do an extra oral exam. For these students the final grade is the maximum between 17 and the simple average of the previous grade and the oral exam.
Formative evaluation is done in or by tasks to be done outside class, to improve the learning process; the elements of formative evaluation will have no weight on the final mark
The evaluation may be either continuous, done through between two and six partial tests and quizzes, done preferably during the classes, weighting 100% of the classification, the number of which is to be defined by the professor who is responsible for the course unit in each academic year, or by a final exam. Students who achieve a grade of 18 or above may have to do an extra oral exam. For these students the final grade is the maximum between 17 and the simple average of the previous grade and the oral exam.
Formative evaluation is done in or by tasks to be done outside class, to improve the learning process; the elements of formative evaluation will have no weight on the final mark
Teaching Staff
- João Miguel Cardoso Dias
- Manuel Baptista Branco [responsible]