MA15110 Algebra 1
Introductory description
This module introduces important algebraic structures including groups, rings and fields. Students will learn how to verify that a set is a group, ring or field, and how to carry out elementary operations in these structures. They will also assimilate permutations, symmetric groups, and alternating groups, and know how to determine the unit group of a ring.
Module aims
The main idea is to teach Algebra “naturally”, similarly to the way children learn schoollevel Algebra. According to this, abstractions should come after calculations. Hence, as far as abstractions are concerned, we are going back to High School. For instance, Isomorphism Theorems, including OrbitStabiliser Theorem, should wait until Algebra3. So is Lagrange’s Theorem (except the abelian groups version that has a quick proof).
Our approach is not to be confused with “Harvard Calculus” from 1990s: we give complete definitions and state theorems precisely in this module.
It is important to get interaction with Foundations. In week 7 the following material is required for Foundations: fields, groups, the multiplicative group, little Fermat theorem, Euler’s totient function. On the other hand, Chinese Remainder Theorem and RSA are to be covered in Foundations.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
 Group Theory: motivating examples (numbers, cyclic group, dihedral group, symmetric group, transformations of the plane), elementary properties, subgroups, Lagrange's Theorem for abelian groups, odd and even permutations.
 Ring Theory: commutative and noncommutative rings, fields, examples (Z[x], Z/nZ, F[x], F[x]/(f)), unit groups, factorisations in Z and polynomials.
 List of covered algebraic definitions: group, subgroup, group homomorphism (including kernel, image, isomorphism), order, sign of permutation, ring, field, subring, ideal, ring homomorphism (including kernel, image, isomorphism), quotient ring
Learning outcomes
By the end of the module, students should be able to:
 understand the abstract definition of a group and a group homomorphism
 be familiar with the dihedral and cyclic groups as well as the group of euclidean transformations of the plane
 perform manipulations with the elements of the symmetric group, representing them as a product of compositions
 understand the orders of elements as well as the proof of Lagrange’s Theorem for abelian groups
 get the working knowledge of the understand the definition of various types of ring, and be familiar with a number of examples, including numbers, polynomials and Z/nZ
 perform manipulations with polynomials over Z, R and C
 learn the unit groups of rings, in particular, of Z/nZ
Indicative reading list
Samir Siksek, Introduction to Abstract Algebra lecture notes,
Ronald Solomon, Abstract Algebra, Brooks/Cole, 2003.
Niels Lauritzen, Concrete Abstract Algebra, Cambridge University Press, 2003
Subject specific skills
This module introduces important algebraic structures including groups, rings and fields. Students will learn how to verify that a set is a group, ring or field, and how to carry out elementary operations in these structures. They will understand the relation between a group, a subgroup and the cosets of a subgroup which leads to Lagrange's theorem. They will also assimilate permutations, symmetric groups, and alternating groups, and know how to determine the unit group of a ring.
Transferable skills
The module reinforces logical thinking and deductive reasoning which are valuable transferable skills. The algebraic structures introduced are the heart of modern cryptography and information security.
Study time
Type  Required 

Lectures  20 sessions of 1 hour (48%) 
Online learning (independent)  9 sessions of 1 hour (21%) 
Private study  13 hours (31%) 
Total  42 hours 
Private study description
Working on assignments, going over lecture notes, text books, exam revision.
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Assessment group D
Weighting  Study time  

Assignments  15%  20 hours 
homeworks 

Inperson Examination  85%  38 hours 
final exam

Assessment group R
Weighting  Study time  

Inperson Examination  Resit  100%  
final resit exam

Feedback on assessment
Marked homework (both assessed and formative) is returned and discussed in smaller classes. Exam feedback is given.
Courses
This module is Core for:
 Year 1 of UMAAG105 Undergraduate Master of Mathematics (with Intercalated Year)

UMAAG100 Undergraduate Mathematics (BSc)
 Year 1 of G100 Mathematics
 Year 1 of G100 Mathematics
 Year 1 of G100 Mathematics

UMAAG103 Undergraduate Mathematics (MMath)
 Year 1 of G100 Mathematics
 Year 1 of G103 Mathematics (MMath)
 Year 1 of G103 Mathematics (MMath)
 Year 1 of UMAAG106 Undergraduate Mathematics (MMath) with Study in Europe
 Year 1 of UMAAG1NC Undergraduate Mathematics and Business Studies
 Year 1 of UMAAG1N2 Undergraduate Mathematics and Business Studies (with Intercalated Year)
 Year 1 of UMAAGL11 Undergraduate Mathematics and Economics
 Year 1 of UECAGL12 Undergraduate Mathematics and Economics (with Intercalated Year)
 Year 1 of UMAAG101 Undergraduate Mathematics with Intercalated Year