F19PL - Abstract Algebra

Course leader(s):

Alexandre Martin (Edinburgh, September)

Aims

The objective of the module is to introduce and develop the concepts and methods of abstract or structural algebra, with emphasis on its generality and widespread applications, and on the efficacy of the axiomatic method and abstract reasoning.

Syllabus

1. Rings and fields (1.1 Subrings of C and abstract rings, 1.2 New examples of rings: polynomial rings and integers modulo n)

2. Euclidean rings (2.1 Introduction to Euclidean rings, 2.2 Euclidean rings of polynomials, 2.3 Irreducible elements, 2.4 The Factorisation Theorem in Euclidean rings)

3. Ring homomorphisms and quotients (3.1 Quotient of polynomial rings, 3.2 Ring homomorphisms, 3.3 Algebraic number fields and finite fields)

4.1 Transformation groups, 4.2 The symmetric group, 4.3 Abstract groups

5. Subgroups and group homomorphisms (5.1 Subgroups and subgroups generated by a family, 5.2 Lagrange's Theorem, 5.3 Group homomorphisms and isomorphisms, 5.4 The "Rank-Nullity" Theorem for groups)

6. Group actions (6.1 Introduction to group actions, 6.2 The Orbit-Stabiliser Theorem, 6.3 Induced actions and applications)

Learning outcomes

By the end of the course, students should be able to do the following:

demonstrate their understanding of the key structures of abstract algebra by proving abstract statements and performing concrete computations.
demonstrate their understanding of the different classes of rings from the course by proving statements about specific rings.
perform computations in quotient rings and determine whether quotients of polynomial rings over a field are fields.
demonstrate their understanding of the notion of group and subgroup by using the key theorems from the course to prove statements about these objects.
demonstrate their understanding of the notion of homomorphism by transferring properties from one structure to another and telling non-isomorphic structures apart.
use the action of a given group on a set to prove statements about a group and its subgroups.
demonstrate transversal mathematical skills: example generation, reading comprehension, knowledge transfer to new settings, clear writing, etc.

Further details

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SCQF Level: 9

Credits: 15