| Code |
14768
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| Year |
2
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| Semester |
S1
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| ECTS Credits |
6
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| Workload |
TP(60H)
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| Scientific area |
Mathematics
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Entry requirements |
N.A.
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Learning outcomes |
General objectives:
To learn, relate and apply concepts and basic results of the integers, group theory and ring theory.
Learning outcomes:
Capacity for abstraction and generalization;
Logical reasoning capacity;
Written and oral communication capacity, using mathematical language;
Capacity for formulating and solving problems concerning algebraic structures.
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Syllabus |
1. Integers
1.1. Divisibility
1.2. Greatest common divisor and least common multiple
1.3. Prime numbers and unique factorization
1.4. Congruences
2. Groups
2.1. Definition and examples
2.2. Subgroups and cosets, Lagrange’s Theorem
2.3. Normal subgroups and quotient groups
2.4. Group homomorphisms and isomorphism theorems
2.5. Cyclic groups
2.6. Permutation groups, Cayley’s Theorem
2.7. Direct product of groups
3. Rings
3.1. Basic definitions and properties
3.2. Examples
3.3. Subrings, ideals and quotient rings
3.4. Ring homomorphisms and isomorphism theorems
3.5. Ring extensions: embedding in a ring with identity and field of quotients
3.6. Polynomial rings in one indeterminate: Euclidean algorithm, divisibility, greatest common divisor, irreducible polynomials, unique factorization
3.7. Ring of Gaussian integers: units, Euclidean division, Gaussian primes, Euclid’s Lemma, unique factorization.
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Main Bibliography |
Fraleigh, J.B. (2003). A First Course in Abstract Algebra (7th edition).
Judson, T.W. (2021). Abstract Algebra: Theory and Applications, http://abstract.ups.edu/index.html.
Malik, D.S., Mordeson, J.N., Sem M.K. (1997). Fundamentals of Abstract Algebra, McGraw Hill,1997.
Monteiro, A.J., Matos, I.T. (2001). Algebra: Um Primeiro Curso (2ª edição), Escolar Editora.
Sobral, M. (1996). Algebra, Universidade Aberta.
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Teaching Methodologies and Assessment Criteria |
Classes integrate theory and practice to enhance students’ understanding. The teacher introduces key concepts, presents and proves fundamental results, and provides examples and applications. Students are encouraged to participate actively by engaging with the teacher and their peers. Independent work is also promoted through solving exercises, guided reading, and problem formulation and solving.
The assessment consists of homework assignments worth 2 marks, and two written tests, each worth 9 marks. Students also have the option to take a final examination, which is marked out of 20.
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Language |
Portuguese. Tutorial support is available in English.
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