# MA151-10 Algebra 1

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

##### 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 school-level 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 Orbit-Stabiliser Theorem, should wait until Algebra-3. 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 1990-s: 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 non-commutative 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

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 (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

homeworks

In-person Examination 85% 38 hours

final exam

##### Assessment group R
Weighting Study time
In-person 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 UMAA-G105 Undergraduate Master of Mathematics (with Intercalated Year)