| Code |
13909
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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.
Competencies to be developed by students:
Capacity for abstraction and generalization;
Logical reasoning capacity;
Written and oral communication capacity, using mathematical language;
Capacity for formulating and solving problems relating to 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 |
Durbin, J. R. (2009) Modern Algebra: An Introduction (6th edition). John Wiley
Fernandes, R. L., Ricou, M. (2004) Introdução à Álgebra, IST Press
Fraleigh, J.B. (2003) A First Course in Abstract Algebra (7th edition), Pearson
Gonçalves, A. (1979) Introdução à Álgebra. Projeto Euclides
Monteiro, A. J., Matos, I. T. (2001) Álgebra: Um Primeiro Curso (2ª edição), Escolar Editora
Goodman, F. M., (2015) Algebra Abstract and Concrete (2.6 edition), SemiSimple Press, Iowa City, IA (Available from http://homepage.divms.uiowa.edu/~goodman/algebrabook.dir/book.2.6.pdf )
Pinter, C.C. (1990). A Book of Abstract Algebra (2nd edition), Dover.
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Language |
Portuguese. Tutorial support is available in English.
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