Occurrence  Teaching cycle 

A  Autumn Term 201819 to Summer Term 201819 
The Pure Mathematics module in Stage 2 focuses on the power of abstraction by developing mathematical theories from axioms in several contexts – Group Theory, Number Theory, Rings and Fields, Geometry. These four themes share many common aspects, progressing in a rigorous manner with a focus on proof, though applications and connections with other areas of mathematics are never far from sight. By the end of the module, students will appreciate the scope and power of pure mathematics, and have a thorough grounding for further study in Stages 3 and 4.
As part of these broad aims, this module introduces students to:
Group Theory (Autumn Term), the study of symmetry through algebra. Symmetry is ubiquitous in mathematics and science, and the mathematical study of symmetry through group theory is one of the most important tools which pure mathematics has brought to the sciences over the past 200 years. By the end of this component students will have encountered some of the most important examples of groups, as well as developing some farreaching theory “from the ground up”, gaining an insight into powerful ideas and learning techniques which are applicable across a wide range of mathematics and science.
Rings and Fields (Spring Term). The study of rings can be seen as a natural generalisation of the study of the integers. These “integerlike” structures crop up all over mathematics and science, and the systematic study of rings has led to applications across modern mathematics. In this component, students will see how really important mathematics can result from the abstraction of familiar objects. Taking familiar operations on numbers such as addition and multiplication and putting them in an abstract context has many farreaching and important mathematical consequences, which this component will explore.
Number Theory (Spring Term), developing the basic theory of numbers in a systematic way. Mathematicians have been studying numbers from the dawn of the subject, and yet there are many basic problems which remain unanswered. Nothing is more fundamental to our view of mathematics than the idea of number – but this is also an area in which students can quickly access deep and difficult problems. The main aim of this component is to answer a problem which is deceptively easy to state – which natural numbers can be expressed exactly as the sum of two squared integers? – but whose solution involves some deep mathematics. Answering this question necessitates the development of some theory which has many other applications giving students a glimpse of the depth and complexity of modern mathematics.
Geometry (Spring/Summer), introducing students to the three “classical" geometries (Euclidean, spherical and hyperbolic) through their concrete models and their isometry groups. Geometry has been an integral part of mathematics since ancient times, but it was not until the 19^{th} century that mathematicians realised that there could be several different types of geometry, depending on the axiomatic framework one starts with. This realisation, and the subsequent formalisation of such notions as distance and curvature, has had profound implications across mathematics and science. In this component, students will gain an insight into the study of geometry using many of the techniques and ideas they have seen in other contexts (notable here is the use of group theory to study isometries). What it means for a geometry to be “curved” will be investigated in two dimensions using the lengths of curves, geodesics and the area of geodesic triangles. For example, by the end of the component students should be able to tell whether they are living in a plane, on a sphere, or on a hyperbolic plane using the GaussBonnet formula.
Studying these four components alongside each other during the course of the year will allow students to see the many connections across different areas of Pure Mathematics; understanding these connections and being able to use ideas and techniques across many contexts is an essential part of the modern mathematician’s toolkit.
Subject content
Group Theory:
Examples: general linear group, symmetric groups, modular groups
Consequences of the group axioms. Order of elements
Symmetric groups: cycle decomposition, decomposition as products of transpositions, parity
Subgroups: special linear groups, alternating groups, cyclic groups, symmetry groups of plane figures; Klein 4group.
Homomorphisms: the concept of a structure preserving map; homomorphisms, structure preserving properties
Isomorphisms: isomorphisms, isomorphic is an equivalence relation, isomorphic groups have the same properties; automorphisms.
Cosets: definition, illustration with examples ; cosets are equal or disjoint; aH=bH b^{1}a in is H
Lagrange's theorem and applications (the order of an element divides the order of a (finite) group)
Fundamental theorem: normal subgroups, quotient groups and homomorphism theorems
Products: external and internal direct products, classification of finite abelian groups
Rings and Fields:
Definition of rings and examples (including matrix rings, rings of functions and polynomial rings)
Consequences of axioms
Subrings and ideals
Fundamental theorem of ring homomorphisms: homomorphisms, isomorphisms, quotient rings, kernels and homomorphism theorems
Special rings: Integral domains, principal ideal domains and fields, including finite fields Z_p
More on ideals: prime and maximal ideals; characterisation of maximal and prime ideals by quotients for commutative R
Division in commutative rings with identity: prime and irreducible elements; unique factorization domains; Euclidean rings. Euclidean rings are PIDs, PIDs are UFDs
Polynomials: for a field F, F[x] is a PID; irreducible polynomials and quotient fields
Construction of multiplication tables for fields of order p^n
Groups of automorphisms of fields with fixed subfield
Number Theory:
Chinese Remainder Theorem. Solving linear congruences, Chinese Remainder Theorem.
Euler's totient function. Fermat's Little Theorem as a special case of Euler's Theorem, with some applications.
Higher order congruences.
Primitive roots. Orders, primitive roots, indices and applications.
Quadratic residues. Legendre symbols, Gauss' lemma and Euler's Criterion.
Quadratic reciprocity.
Sum of two squares. Fermat's method of descent.
Arithmetic and multiplicative functions.
Geometry:
Euclidean space.
Length and angle in Euclidean space; the metric properties of distance.
Rotations, reflections and translations (in 2 and 3 dimensions). How these sit inside the full group of Euclidean isometries. Generating O(3) by reflections.
What makes a straight line straight? The “shortest path” approach to distance measurement. The “infinitesimal length element” (Riemannian metric) in the plane.
Spherical space.
Segments of spherical lines (great circles) define distance on a sphere. The corresponding Riemannian metric in exterior and interior coordinate (spherical polar) form.
Spherical triangles and their trigonometry.
The isometry group O(3). Its finite subgroups and their classification.
Spherical line segments and shortest paths.
Spherical triangles and their GaussBonnet formula.
Hyperbolic space.
The Poincare models: upper halfplane and open unit disc. Length via Riemannian metrics in each. The isometry between the two. PSL(2;R) and PSU(1; 1) as respective isometry groups and their conjugacy in the full group of Mobius transformations PSL(2;C). The “ideal boundary at infinity". Types of isometries: elliptic, parabolic and loxodromic/hyperbolic.
Hyperbolic lines. Between any two points there is a unique hyperbolic line; the isometry group acts transitively on hyperbolic lines. Hyperbolic line segments are length minimisers (geodesics).
Geodesic triangles and the GaussBonnet formula.
Academic and graduate skills
During the course of this module students will develop their ability to reason in a rigorous, precise and logical manner, proceeding from a short list of axioms to a wealth of important mathematics.
Maths graduates are characterised by their ability to think precisely and logically, to use reason and to be able to justify what they assert because it is built on a solid footing. This module helps develop these essential skills.
The module also looks outwards to applications. Students will learn material and techniques with a wide range of applications outside pure mathematics: for example, group theory has applications across including physics, chemistry and biology; ring theory and number theory have many applications in the digital age, from computing through codes and cryptography to digital signal processing; the development of geometry has had an immense impact on the way we view the universe, from the very large scale to the very small scale, from cosmology to materials science.
Task  Length  % of module mark 

University  closed examination Pure Mathematics: Group Theory 
1.5 hours  25 
University  closed examination Pure Mathematics: Number Theory 
1.5 hours  25 
University  closed examination Pure Mathematics: Rings & Fields and Geometry 
3 hours  50 
None
Students only resit components which they have failed.
Task  Length  % of module mark 

University  closed examination Pure Mathematics: Group Theory 
1.5 hours  25 
University  closed examination Pure Mathematics: Number Theory 
1.5 hours  25 
University  closed examination Pure Mathematics: Rings & Fields and Geometry 
3 hours  50 
Current Department policy on feedback is available in the undergraduate student handbook. Coursework and examinations will be marked and returned in accordance with this policy.
