# MA268-10 Algebra 3

Department
Warwick Mathematics Institute
Level
Samir Siksek
Credit value
10
Module duration
10 weeks
Assessment
Multiple
Study location
University of Warwick main campus, Coventry

##### Introductory description

This is a second abstract algebra module for Mathematics students.

##### Module aims

It is a second Abstract Algebra module, where the students should get workable knowledge of many algebra concepts. Compare to joint degree students, doing Groups and Rings, the students will get extended knowledge of several topics.

##### 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: 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.
##### Learning outcomes

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

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

##### Subject specific skills

Students will improve their skills in thinking algebraically in a variety of settings. This includes working with axiomatic definitions of algebraic objects and analysing the structure and relationships between algebraic objects using fundamental tools such as subobjects and homomorphisms, laying a foundation for future study in algebra, number theory and algebraic geometry.

##### Transferable skills

The module emphasises the power of generalisation and abstraction. Students will improve their ability to analyse abstract concepts and to solve problems by selecting and applying appropriate abstract tools.

## Study time

Type Required
Lectures 20 sessions of 1 hour (20%)
Online learning (independent) 9 sessions of 1 hour (9%)
Private study 13 hours (13%)
Assessment 58 hours (58%)
Total 100 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
Examination 85% 38 hours
##### Assessment group R
Weighting Study time
In-person Examination - Resit 100%
##### 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 2 of UMAA-G105 Undergraduate Master of Mathematics (with Intercalated Year)
• Year 2 of G100 Mathematics
• Year 2 of G100 Mathematics
• Year 2 of G100 Mathematics