# MA268 Algebra 3

**Lecturer: **Samir Siksek

**Term(s):** Term 1

**Status for Mathematics students:** Core for Maths

**Commitment:** 30 one-hour lectures plus assignments

**Assessment:** 85% by 2-hour examination, 15% coursework

**Formal registration prerequisites: **None

**Assumed knowledge: **

- Number theory: prime factorisation, Euclidean algorithm, gcd and lcm, Chinese Remainder Theorem
- Sets and functions: basic set theory, injective and surjective functions, bijections and their inverses, relations

- Binary operations
- Fermat's Little Theorem, Euler's Theorem
- Polynomials: multiplication and division, Euclidean algorithm, Remainder Theorem
- Permutations: multiplication, decomposing into disjoint cycles, transpositions, even and odd permutations
- Group theory: groups, abelian groups, cyclic groups, subgroups, key examples (numbers, dihedral group, isometry groups, symmetric group, alternating group), order of an element, Lagrange's Theorem
- Ring theory: definitions of rings, subrings, ideals, cosets of ideals, quotient rings, fields. Key examples ($\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}/n\mathbb{Z}$, matrix rings, polynomial rings).

- Matrices, determinants, echelon form, elementary matrices
- Vector spaces, subspaces, bases, span, linear independence
- Linear transformations

**Synergies:**

**Leads to: **The following modules have this module listed as **assumed knowledge** or **useful background:**

- MA257 Introduction to Number Theory
- MA3E1 Groups and Representations
- MA3G6 Commutative Algebra
- MA3J9 Historical Challenges in Mathematics
- MA377 Rings and Modules
- MA3F1 Introduction to Topology
- MA3K4 Introduction to Group Theory
- MA3J3 Bifurcations, Catastrophes and Symmetry
- MA3D5 Galois Theory
- MA3H6 Algebraic Topology
- MA3J2 Combinatorics II
- MA3A6 Algebraic Number Theory
- MA4L6 Analytic Number Theory
- MA4H4 Geometric Group Theory
- MA426 Elliptic Curves
- MA473 Reflection Groups
- MA453 Lie Algebras
- MA4J8 Commutative Algebra II
- MA4M6 Category Theory

**Aims**:

This is a second course on groups, rings and fields that delves deeper into those topics. We aim to attain a better understanding of groups both as abstract objects and through their actions on sets. We also want to understand how unique factorization can be generalized in rings that have a Euclidean algorithm.

**Content**:

- Group Theory: quaternionic group, matrix group, coset, Lagrange’s theorem, quotient group, isomorphism theorem, free group, group given by generators and relations, group action, G-set G/H, orbit, stabiliser, the orbit-stabiliser theorem, conjugacy class, classes in S_n, classification of groups up to order 8.
- Ring Theory: domain, isomorphism theorem, Chinese remainder theorem for Z/nZ and F[x]/(f), unit group, prime and irreducible element, factorization, Euclidean domain, characteristic of a field, unique factorization domain, ED is UFD, finite subgroup of units in fields.
- Module Theory: module, free module, internal and external direct sum, free abelian group, unimodular Smith normal form, the fundamental theorem of finitely generated abelian groups.
- List of covered algebraic definitions: direct product, coset, normal subgroup, quotient group, ideal, quotient ring, domain, irreducible element, prime element, euclidean domain, unique factorisation domain, direct product, free group, generators and relations, module, free module, direct sum, unimodular Smith normal form, action, orbit, stabiliser, fixed points.

**Objectives**: By the end of the module, students should be able to:

- have a working knowledge of the main constructions and concepts of theories of groups and rings
- recognise, classify and construct examples of groups and rings with specified properties by applying the algebraic concepts

**Books: **

- Ronald Solomon, Abstract Algebra, Brooks/Cole, 2003.
- Niels Lauritzen, Concrete Abstract Algebra, Cambridge University Press, 2003
- John B. Fraleigh, A first course in abstract algebra, Pearson, 2002
- Joseph A. Gallian, Contemporary Abstract Algebra, Cengage Learning, 2012