Algebra II

Code	14777
Year	2
Semester	S2
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	N.A.
Learning outcomes	General Objectives: Assimilate, relate, and apply concepts and results in Group Theory, Ring Theory, and Field Theory. Build upon the elementary aspects of group theory and ring theory covered in Algebra I, where various concrete examples were presented. Provide students with the opportunity to deepen their understanding of abstract mathematical reasoning and proofs initiated in the Algebra I course. Competencies to be Developed by Students: Capacity for abstraction and generalization. Logical reasoning skills. Proficiency in written and oral communication using mathematical language. Ability to formulate and solve problems related to algebraic structures.
Syllabus	1. Groups 1.1. Group Actions 1.2. Burnside's Theorem 1.3. Sylow's 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 Splitting field of a polynomial 3.1.3 Algebraic and transcendental elements 3.1.4 Ruler and compass constructions 3.2. Galois Theory 3.2.1 The Galois group 3.2.2 Normal and separable extensions 3.2.3 The Galois correspondence 3.2.4 Solvability of polynomial equations
Main Bibliography	Dummit, David S.; Foote, Richard M., Abstract algebra. Third edition. John Wiley & Sons, Inc., Hoboken, NJ, 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
Teaching Methodologies and Assessment Criteria	The classes combine theory with practice, where the teacher introduces concepts, states and proves fundamental results, and offers examples and applications. Students are not only encouraged to actively participate in classes and interact with their teacher and colleagues but also to work independently by solving exercises, engaging in guided reading, and practicing problem formulation and solving. The evaluation conducted during the teaching-learning process includes homework problems and two written tests. Moreover, students have the option to take a final examination.
Language	Portuguese. Tutorial support is available in English.