Module MATH2781: Algebra II

Department: Mathematical Sciences

MATH2781: Algebra II

Prerequisites

• One of:
• Calculus I (Maths Hons) (MATH1081) OR Calculus I (MATH1061)
• AND one of:
• Linear Algebra I (Maths Hons) (MATH1091) OR Linear Algebra I (MATH1071)

Corequisites

• None

Excluded Combination of Modules

• None

Aims

• 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.

Content

• Rings and fields:
• Rings, subrings and fields. First examples (and non-examples) of rings and fields. Definition of a ring. The integers modulo n. Subrings and Fields.
• Integral domains, units, polynomial rings. Zero divisors, Examples and non-examples of integral domains. The group of units in a ring. Units of Z/n. Polynomials over a field and the division algorithm.
• Divisibility and the greatest common divisor in a ring. The notions of divisibility and of greatest common divisor (gcd), The Euclidean algorithm, Existence of gcd for F[x] and Z.
• Factorization in rings. Irreducible polynomials in F[x], Basic Criteria for Irreducibility, Gauss’ Lemma, Eisenstein’s Criterion, Unique Factorization in F[x], prime and irreducible elements, Unique Factorization Domains (UFDs).
• Homomorphisms. Definitions and properties, Examples of ring homomorphisms, Product of rings and projections.
• Ideals and quotient rings. Basic definitions and examples, Kernels of homomorphisms, Quotients of rings by ideals, Equivalence relations on a ring modulo an ideal, The quotient map and the First Isomorphism Theorem (FIT). Prime and maximal ideals.
• Principal Ideal Domains (PID). Definitions and Examples, Z and F[x] and PIDs, PIDs are UFDs, Fields as quotients, Finite Fields, The Chinese Remainder Theorem (CRT) for PIDs.
• Groups:
• The definition of group. Examples of groups: Cyclic and dihedral groups (composition of symmetries, generators and relations), symmetric and alternating groups, matrix groups.
• Subgroups and cosets. Order of an element of a group, cosets of a group with respect to a subgroup, Lagrange's Theorem, Euler’s and Fermat’s theorems in number theory.
• Group homomorphisms, kernel and image, isomorphisms. The notion of quotient groups, normal subgroups, the First Isomorphism Theorem. Direct products. Isomorphisms between various groups. List of groups of small order. Simple groups.
• Group actions: Action of a group on a set. Orbits, stabilizers and the orbit-stabilizer theorem, Cayley's theorem. Cauchy's theorem, classification of groups of order twice a prime.
• Conjugacy and classification. Conjugate elements, conjugacy classes in Sn. Centre of a group; classification of groups of order p2.
• Cyclic and finitely generated abelian groups: Classification and structure.

Learning Outcomes

• By the end of the module students will:
• Be able to solve seen and unseen problems on the given topics.
• Have an awareness of the abstract concepts of theoretical mathematics in the field of Algebra.
• Have a knowledge and understanding of fundamental theories of these subjects demonstrated through one or more of the following topic areas: Rings and fields, example of groups, generators, homomorphisms. Group actions, Equivalence relations. Structure of finitely generated abelian groups.

• In addition students will have the ability to undertake and defend the use of alternative mathematical skills in the following areas with minimal guidance: Abstract reasoning.

• Students will have basic mathematical skills in the following areas: abstract reasoning, problem solving.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Problem classes show how to solve example problems in an ideal way, revealing also the thought processes behind such solutions.
• Tutorials provide active problem-solving engagement and immediate feedback to the learning process.
• Formative assessments provide feedback to guide students in the correct development of their knowledge and skills in preparation for the summative assessment.
• Summative assignments test achievement of learning outcomes and provide instant feedback to students about their mastery of the topics.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

There will be at most one formative or summative assignment each week.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University