Subject: Computational Algebra

Scientific Area:

Mathematics

80 Hours

Number of ECTS:

7,5 ECTS

Language:

Portuguese

# Overall objectives:

O1 - To master fundamental concepts and results within the integers ring, the polynomial rings in a single variable and also in several variables.
O2 - To identify points of similarity between the Integers ring and the polynomial rings in a single variable over a field.
O3 - To gain theoretical and practical knowledge which form part of the basis of subjects like Commutative Algebra, Algebraic Geometry and Algebraic Number Theory.
O4 - Knowledge of how to use a Computer Algebra System to construct examples and to solve problems.
O5 - Acquisition of research, writing and communication skills.

# Syllabus:

1 - The Integers: The Greatest Common Divisor and the Least Common Multiple. Euclides Algorithm. Bezout's lemma. Factorization: The Fundamental Theorem of Arithmetic. Ideals. Congruences. Quotient rings. Principal Ideal Domains. Euclidean Domains.
2 - Polynomial rings in a single variable: Formal series and Polynomials. Polynomial Functions. Division Algorithm. The Fundamental Theorem of Algebra. Factorization: irreducible polynomials and Unique Factorization Domains.
3 - Galois Theory: Field Extensions. Galois Groups. Galois Correspondence. Solution by radicals of polynomial equations.
4 - Polynomial rings in several variables: Monomial orders. Division Algorithm. Hilbert's Base Theorem. Gröbner Bases and Buchberger's Algorithm. Hilbert's Nullstellensatz and systems of polynomial equations.

# Literature/Sources:

Thomas W. Hungerford , Algebra. , Springer Verlag
R.L. Fernandes, M. Ricou , 2004 , Introdução à Álgebra , IST Press
David Cox, John Little, Donal O'Shea , 2006 , Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commuta , Springer-Verlag
Ian Stweart , 1989 , Galois Theory , Chapman and Hall

# Assesssment methods and criteria:

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
The lectures are used to introduce the concepts, results and methods to the student. Also, lecture notes will be provided by the teacher. In the tutorial classes, the student is expected to solve problems from the exercise sheets given by the teacher, individually or in a group. These classes will be held in computer room were the students will make use of a Computer Algebra System (e.g., SAGE) for problem solving. The evaluation will be done through the realization of two tests, both with a weight of 35% of the final grade, and a coursework, which is to be handed in the end of the semester and accounts for 30% of the final grade. The coursework should be written in the form of a scientific article and will be subject to an oral presentation.