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Department of Mathematical Sciences

# MATH2581 Algebra II

Algebra forms the basis for much of mathematics and is a prerequisite for several third and fourth year modules such as Algebraic Geometry, Galois Theory, Geometry, Number Theory, Representation Theory, and Topology.

### Term 1

The first part of the module is an introduction to the theory of rings, which can be thought of as systems of mathematical objects which can be added, subtracted and multiplied, but not necessarily divided (in a field we can also divide). We look at various examples of rings and fields: integers, rationals, complex numbers, integers modulo n, polynomials etc. Among the topics covered are various properties of polynomials over fields (unique factorisation and irreducibility, for example), zero divisors and integral domains, as well as homomorphisms between rings, ideals and quotient rings. In the last few lectures we revisit vector spaces and consider so-called modules which are generalisations of vector spaces, where the scalars come from a ring rather than a field.

### Term 2

In the second part we investigate the notion of a group, a concept that is highly relevant in many branches of mathematics and physics. Important themes are those of homomorphisms, normal subgroups and quotient groups. One central idea is that of a group acting on, or permutin the elements of a set. For example, the group of rotational symmetries of a cube acts on the set of edges of the cube (each rotation permutes the edges). By exploiting this idea we can obtain classification results in group theory (a partial converse to Lagrange's theorem, description of all groups of order 2p and p2 for a prime p). We then turn to the description of finitely generated abelian groups showing that each is isomorphic to the direct product of cyclic groups.

## Outline of Course

Aim: To introduce further concepts in abstract algebra and linear algebra, develop their theory, and apply them to solve problems in number theory and other areas.

### Term 1

• Rings and fields (16 lectures): Definitions and examples of rings e.g. ℤ/nℤ, matrix rings, quaternions, polynomial rings. Homomorphism of rings, integral domains and fields, units of a ring. Polynomials over a field, greatest common divisor of polynomials, division algorithm, Euclidean algorithm, irreducibility: Gauss's lemma, Eisenstein's criterion. Ideals and Quotient Rings: Ideals e.g. kernels of ring homomorphisms, quotient rings, first isomorphism theorem for rings, Chinese remainder theorem for rings, prime/maximal ideals and their characterisation in terms of quotient rings.
• Group theory I (4 lectures): Review of formal definition of group and elementary manipulations. Notion of subgroup, order of an element of a group, cosets of a group, Lagrange's Theorem. Examples of groups: Cyclic and dihedral groups, rotational symmetry groups of regular polyhedra, matrix groups, symmetric and alternating groups. Group homomorphism, kernel and image, isomorphism. Operations on groups: Direct product, product of cyclic groups. Isomorphisms between various groups including groups of platonic solids, Cayley's theorem.

### Term 2

• Group theory II (12 lectures): Homomorphisms and Quotient Groups: group homomorphism, Normal subgroups. First isomorphism theorem for groups and applications. Automorphisms, inner automorphisms. Simple groups: An for n≥5. Commutator subgroup. Group actions: Action of a group on a set. Orbits, stabilisers and the orbit-stabiliser theorem. Cauchy's theorem. Conjugacy, conjugacy classes in Sn. Centre, centre of a p-group, groups of order p2. Sylow theorems. Finite subgroups of O(2) and SO(3). Finitely generated abelian groups: Classification, uniqueness of rank and torsion coefficients. Systems of linear equations in integers, recognition of groups from a finite presentation.
• Topics in linear algebra (6 lectures): Vector spaces over arbitrary fields. Dual spaces and bases, quotient vector spaces, isomorphism theorems. Introduction to modules, structure theorem for finitely generated modules over a PID (statement), Jordan decomposition revisited.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.