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	Assessment during the teaching and learning period will consist of homework assignments and two written tests. Students may also, additionally, write a final examination. Homework assignments will be worth 1 mark. The first written test will be marked out of 9 and the second out of 10, each with a duration of two hours. The final mark (FM) for the teaching and learning process will be calculated according to the following formula: FM = T + P1 + P2, where T represents the homework mark, and P1 and P2 represent the marks obtained in the first and second written tests, respectively. Students will be admitted to the examination if they obtain a minimum final mark of 5 (after rounding) or if they have attended at least 70% of the scheduled classes. Students will pass the course if they obtain a final mark of 10 or higher (after rounding). The final examination will be worth 20 marks. Students with special status have their own rules set out in the Academic Regulations.
Language	Portuguese. Tutorial support is available in English.