Algebra II

Code	13918
Year	2
Semester	S2
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	N.A.
Learning outcomes	General objectives: To assimilate, relate and apply concepts and results on Group Theory, Ring Theory and Field Theory. The most elementary aspects of ring theory have already been studied in Algebra I. Some examples of rings have already been studied, such as the ring of Gauss integers or the ring of Polynomials over a Field. In the course of Algebra II we make a study more abstract and general not limiting us to concrete examples. We will study, for example, the ring of integers in an algebraic number field as well as function fields over a finite field. 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.
Syllabus	1. Groups 1.1. Group Actions 1.2. Burnside's Theorem 1.3. Sylow theorems 1.4. Free Abelian Groups 1.5. Finitely generated Abelian groups 1.6. Finite Abelian Groups 1.6.1 Decomposition in p-groups 1.6.2 Decomposition of p-groups 1.6.3 The Fundamental Theorem of Finite Abelian Groups 2. Rings 2.1. Euclidean Domains 2.2. Principal ideals domains 2.3. Unique factorization domains 3. Fields 3.1. Field extensions 3.1.1 Generalities 3.1.2 Rupture field of a polynomial 3.1.3 Algebraic and transcendent elements 3.1.4 Constructions with ruler and compass 3.2. Galois Theory 3.2.1 The Galois group 3.2.2 Normal and separable extensions 3.2.3 The correspondence of Galois 3.2.4 Resolution of equations by means of radicals
Main Bibliography	Dummit, David S.; Foote, Richard M., Abstract algebra. Third edition. John Wiley & Sons, Inc., Hoboken, NJ, 2004. Fernandes, R. L., Ricou, M., Introdução à Algebra, IST Press, 2004. Fraleigh, J.B. A First Course in Abstract Algebra (7th edition), Pearson, 2003 Milne, J.S., Group Theory and Fields and Galois Theory, 2012 (Available from http://www.jmilne.org/math/CourseNotes/FTe6.pdf ) Monteiro, A. J., Matos, I. T., Álgebra: Um Primeiro Curso (2ª edição), Escolar Editora, 2001 Spindler, Karlheinz, Abstract algebra with applications. Vol. II. Rings and Fields. Marcel Dekker, Inc., New York, 1994 Stewart, I, Galois Theory, 4ed, CRC Press, 2015 Tignol, J. P, Galois' Theory of Algebraic Equations, World Scientific, 2001
Language	Portuguese. Tutorial support is available in English.