Algebra holds a unique and foundational place in mathematics, developed extensively from an early stage. At its core, this area of mathematics studies algebraic structures—sets equipped with one or more operations. This course focuses on the essential pillars of abstract algebra: groups, rings, fields, and modules. Modern algebra is a vibrant and dynamic field, with powerful techniques and results that are integral to virtually every area of mathematics. Moreover, its concepts find wide-ranging applications in physics, engineering, and computer science, making the study of algebra both fundamental and highly relevant today.
The course is designed to foster a collaborative theorem-proving approach. Several days before each class, you will receive lecture notes containing definitions, examples, and propositions to be proved. During class, we will engage in discussions around your approaches, insights, and questions to deepen your understanding and facilitate active participation.
Prerequisites
There are no formal prerequisites, although familiarity with matrices or, more broadly, linear algebra will enrich your experience by expanding the range of examples and connections.
Curriculum
- Properties of division in integers. The GCD and Euclidean algorithm in \( \mathbb{Z} \). Bézout’s identity. Factoring into primes. Unique factorization in \( \mathbb{Z} \).
- Definition and examples of groups. Abelian and non-abelian groups. Cayley diagrams.
- Symmetric groups \( S_n \). Cycle notation.
- Dihedral groups \( D_n \).
- Definition of a subgroup. The center of a group \( Z(G) \) and the derived subgroup \( G’ \) as examples of subgroups.
- Order of an element of a group. Cyclic groups. Classification of subgroups of cyclic groups.
- Cosets, Lagrange’s theorem.
- Group homomorphism. Kernel and image of a homomorphism. Isomorphism of groups.
- Group action. Orbits, stabilizers. Cayley’s Theorem.
- Inner automorphism. Conjugacy classes.
- Sign of a permutation. Alternating Groups \( A_n \).
- Normal subgroups. Quotient groups.
- The first isomorphism theorem. Other isomorphism theorems.
- Free groups. Groups presented by generators and relations.
- * The Chinese remainder theorem.
- * The fundamental theorem of finitely generated abelian groups.
- Definitions of a ring and of a rng (ring without unity). Subrings.
- Integral domains. Fields. Characteristic of a ring. Examples of finite fields. Fermat’s little theorem.
- Ideals. Quotient ring. Prime and maximal ideals. Principal ideals.
- Principal ideal domains. Divisibility and factorization in Euclidean domains.
- Ring of polynomials. Ideals of a polynomial ring. Irreducible polynomials. Eisenstein’s irreducibility criterion.
- Unique factorization in \( \mathbb{Z}[x] \).
- Modules over a ring and their properties.
- * Algebras, division algebras. The algebra of quaternions \( \mathbb{H} \). Frobenius theorem classifying finite-dimensional real division algebras.
- * Extensions of fields.
- * Symmetric polynomials. Fundamental theorem of algebra.
* – if time allows.
Grading & Assessment
There will be two classes each week: one lecture and one problem-solving class. The lecture is not a traditional presentation; it is a guided discussion of the pre‑read notes where we jointly prove the day’s theorems. We will use a collaborative online whiteboard (Miro) for each session where lecture notes or problem sets are uploaded in advance. Each week you should write solutions to at least four problems total across those boards. You will receive 1 point for each correct solution, up to a maximum of 4 points per week. Additional bonus points may be awarded for the shortest and most elegant solutions or for solutions to the most challenging exercises.
Both the midterm and final exams are oral, open-book, and open-notes. The midterm consists of 5 problems worth 3 points each, and the final exam consists of 5 problems worth 4 points each. You also receive 1 bonus point each time you are the first to correct the professor’s error during a lecture.
Grade = (regular points earned + bonus points) / (regular maximum possible points). Thus it is possible to score above 100%. To receive an A+, you need to earn at least 90% of regular points; an A requires 85%, and so on.
Textbooks
- D.Dummit, R.Foote, Abstract Algebra.
- J.Fraleigh, A First Course in Abstract Algebra.
- J.Gallian, Contemporary Abstract Algebra.
- C.Pinter, A Book of Abstract Algebra.
Teasers and Spoilers
- The order of every element of a finite group divides the order of this group.
- Every finite group is isomorphic to a subgroup of a symmetric group.
- For every integer \( n \) and a prime number \( p \) we have that \( p \) divides \( n^p-n \).
- Every ideal of the ring \( \mathbb{R}[x] \) is principal, i.e. it is generated by one element.
- There exists a unique (up to an isomorphism) finite-dimensional division algebra over \( \mathbb{R} \) that is not a field.