General Information

Course: Algebra 3
Language: English
ECTS credits: 7
Number of hours: 84
Teacher: Lucas GNECCO HEREDIA
Contact information: lucas.gneccoheredia@psl.eu

Meeting times

6 hours per week, on Tuesdays. The course is organized in the following blocks:

Lecture: Tuesday, 09h00 - 12h15
Tutorials: Tuesday, 13h45 - 17h00

Syllabus (course description)

This course is an introduction to abstract algebra, with a focus on the structures that organize mathematical reasoning across algebra, geometry, symmetry, and computation. Students will study the fundamental properties of integers, modular arithmetic and groups. At the end of the course, there will be an introduction to rings and polynomial rings. Students will learn how these ideas are used to understand symmetry, structure, and algebraic operations.

This course provides the foundation for more advanced work in algebra, number theory, coding theory, cryptography, and computational mathematics. It also serves as a bridge between linear algebra and more abstract mathematical structures.

Learning Objectives

Foundations and Number Theory

Work with integers, divisibility, gcd, lcm, and the fundamental theorem of arithmetic.
Understand modular arithmetic and the role of congruence classes.
Use induction and well-ordering principles in mathematical arguments.
Reason rigorously with equivalence relations and functions.

Groups and Symmetry

Define and work with groups, subgroups, cyclic groups, and permutation groups.
Compute orders of elements and groups, and understand finite group structure.
Analyze isomorphisms, cosets, and Lagrange’s theorem.
Understand normal subgroups and quotient groups.
Work with homomorphisms and the First Isomorphism Theorem.

Rings and Algebraic Structures

Define and analyze rings, integral domains, ideals, and quotient rings.
Understand ring homomorphisms and the relationship between substructures and quotient structures.
Work with polynomial rings and basic properties of polynomial arithmetic.
Relate abstract algebraic concepts to concrete algebraic computations and examples.

Proof and Structure

Construct clear and rigorous proofs in abstract algebra.
Distinguish among definitions, examples, theorems, and corollaries.
Use structural reasoning to justify algebraic results.
Explain mathematical arguments in a logically precise and well-organized way.

Applications

Recognize how algebraic structures arise in symmetry, transformations, and computational settings.
Appreciate applications to coding theory, cryptography, and symbolic computation.
Connect abstract algebra to the broader mathematical curriculum and to modern computational problems.

Books and references

The main textbook for this course is:

[Gallian] Joseph A. Gallian. Contemporary Abstract Algebra (7th edition). Brooks/Cole, Cengage Learning.

Additional references:

Serge Lang and Frank W. Gehring. Undergraduate Algebra.

The course follows the Gallian-based plan, with emphasis on the core concepts of group theory and elementary ring theory.

Evaluation

There will be one final exam and regular quizzes during the semester. The final grade will be computed as follows:

Evaluation	Weight
Final exam	80%
Quizzes	20%

All quizzes will have equal weight, and will be announced in advance via Moodle or via email.

Calendar

The pace may be adjusted depending on the needs of the class, but the overall structure is fixed.

Date	Session	Topic	Reading
Tue, 15-09-26	1.1	Course intro (20 min).Properties of integers. Well Ordering Principle and relation with Induction. Division algorithm. Greatest common divisor and Relatively Prime Integers. Fundamental theorem of arithmetic. Least Common Multiple.	Gallian, Chapter 0
Tue, 15-09-26	1.2	Modular arithmetic and examples. Mathematical induction. Equivalence relations, partitions. Functions	Gallian, Chapter 0
Tue, 22-09-26	2.1	Groups, examples of groups, basic properties and concepts	Gallian, Part 1, Chapter 1: Introduction to groups. Chapter 2: Groups
Tue, 22-09-26	2.2	Continuation
Tue, 29-09-26	3.1	Finite groups: order of a group, order of an element. Subgroups. Center of a subgroup.	Gallian, Part 1, Chapter 3: Finite groups; Subgroups
Tue, 29-09-26	3.2	Continuation
Tue, 06-10-26	4.1	Mock Exam # 1, Chapters 0 to 3
Tue, 06-10-26	4.2	Cyclic groups: properties and important results	Gallian, Part 1, Chapter 4: Cyclic groups
Tue, 13-10-26	5.1	Continuation: Classification of subgroups of a cyclic subgroup.
Tue, 13-10-26	5.2	Continuation
Tue, 20-10-26	6.1	Permutation groups: definition, examples and basic properties	Gallian, Part 1, Chapter 5: Permutation groups
Tue, 20-10-26	6.2	Continuation
Tue, 03-11-26	7.1	Mock Exam # 2, Chapters 4 and 5
Tue, 03-11-26	7.2	Isomorphisms: definition and examples, Cayley's Theorem, automorphisms.	Gallian, Part 1, Chapter 6: Isomorphisms
Tue, 10-11-26	8.1	Cosets, Lagrange's Theorem,	Gallian, Part 1, Chapter 7: Cosets and Lagrange's Theorem
Tue, 10-11-26	8.2	Continuation
Tue, 17-11-26	9.1	External Direct Products: definition, important results and examples. Application to real life problems	Gallian, Part 1, Chapter 8: External Direct Products
Tue, 17-11-26	9.2	Continuation
Tue, 24-11-26	10.1	Mock Exam # 3, Chapters 6 to 8
Tue, 24-11-26	10.2	Normal subgroups and Factor groups	Gallian, Part 1, Chapter 9: Normal subgroups and Factor groups
Tue, 01-12-26	11.1	Homomorphism, First Isomorphism Theorem, Normal subgroups and kernels of homomorphisms.	Gallian, Part 1, Chapter 10: Group Homomorphisms
Tue, 01-12-26	11.2	Continuation. Mention the Fundamental theorem of Finite Abelian Groups.
Tue, 08-12-26	12.1	Rings and integral domains	Gallian, Part 3, Chapter 12: Introduction to rings. Chapter 13: Integral domains
Tue, 08-12-26	12.2	Continuation
Tue, 15-12-26	13.1	Ideals and Factor rings.	Gallian, Part 3, Chapter 14: Idela and factor rings.
Tue, 15-12-26	13.2	Ring homomorphisms	Gallian, Part 3, Chapter 15: Ring homomorphisms
Tue, 12-01-27	14.1	Mock Exam # 4, Chapters 9 to 15	Gallian, Part 3, Chapter 16: Polynomial rings
Tue, 12-01-27	14.2	Review session: course roadmap