Dr. Thomas Keilen Algebra I - MA 242

#### Literature:

 Thomas Keilen, Algebra I. Lecture notes, University of Warwick, 2002. John Humphreys, A Course in Group Theory, Oxford. P M Cohn, Algebra, Vol. 1, Wiley. I N Herstein, Topics in Algebra, Wiley.

#### Assingments and Solutions:

Post Script Files: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 .
Latex Files: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 .
Style Files: ag.sty , tom.sty .

#### Marking Scheme for Assessed Home Work in Algebra I

The exercises may consist of one or more parts. The maximum number of marks for each part of an exercise is four. Please assign them according to the following scheme:
4 Marks
The solution is (nearly) completely correct - some small error in a calculation (or similar) should not matter.
3 Marks
The main ideas are correct, even though some details are wrong.
2 Marks
Never - unless you are absolutely undecided.
1 Marks
The solution is basically wrong, even though it contains some interesting idea.
0 Marks
The solution is hopelessly incorrect.
The main decision which has to be taken is, whether the solution belongs rather to the category "correct" or "incorrect".

Assessed Home Work
Set 1Exercise 24 marks
Set 2Exercise 14 marks
Set 3Exercise 1 a.4 marks
Exercise 1 b.4 marks
Exercise 2 a.4 marks
Exercise 2 b.4 marks
Set 4Exercise 1 a.4 marks
Exercise 1 b.4 marks
Exercise 24 marks
Set 5Exercise 1 or 34 marks
Set 6Exercise 1 or 3 a./b.4 or 4+4=8 marks
Set 7Exercise 3 a.4 marks
Exercise 3 b.4 marks
Set 8Exercise 14 marks

Maximum Number of Marks60 marks

Status: Core for Maths.

Assessment: Assignments (15%), two-hour examination (85%).
This module will be examined in week 21, the first week of Term 3.

Commitment: 30 one-hour lectures plus four assignments.

Prerequisites: MA129 Foundations, MA106 Linear Algebra, and MA130 From Geometry to Groups are an advantage but by no means essential. We will recap all first year material in the module.

Content: Group theory takes up about half of Algebra I. We study in particular groups, subgroups and homomorphisms from one group to another, with many examples of each idea. Groups occur in nature as the symmetry groups of geometric or other mathematical or physical constructions; we treat the notion of an abstract group G acting on a set S. A key structural element of algebra introduced in this module is the notion of a normal subgroup H of a group G and the associated quotient group G/H. We will also study group actions. These have many applications including Sylow's theorem, which we shall see is in some sense a partial converse to Lagrange's theorem.

We next study quadratic forms. A quadratic form is a homogeneous quadratic polynomial expression in several variables. Quadratic forms occur in geometry as the equation of a quadratic cone, or as the leading term of the equation of a plane conic or a quadric hypersurface. By a change of coordinates, we can always write q(x) in the diagonal form. For a quadratic form over R, the number of positive or negative diagonal coefficients ai is an invariant of the quadratic form which is very important in applications.

We discuss a square matrix matrix M as an endomorphism of a vector space V. We study Jordan canonical form of 2x2 and 3x3 matrices. The general case will be treated in MA245 Algebra II.

Aims: To provide a further introduction to abstract group theory, building upon the material in year 1 from Foundations and taking in some of the classical theorems on finite groups.

To develop upon first year linear algebra, paying particular attention to canonical forms of linear maps, matrices and bilinear forms.

To make students familiar with some important techniques in linear algebra and group theory which are used in other modules.

Objectives: By the end of the module students should be familiar with: the isomorphism theorems for groups and applications; quotient groups; Cayley's theorem; group actions and lots of applications, including the class equation and Sylow's theorem; the theory and computations of the the Jordan normal form of matrices and linear maps; bilinear forms, quadratic forms, and choosing canonical bases for these.

Leads to: MA245 Algebra II, third year algebra modules, in particular MA3D5 Galois Theory and MA442 Group Theory. Some of the theory is also needed in MA371 Qualitative Theory of ODEs.

Universität KaiserslauternFachbereich MathematikArbeitsgruppe Algebraische Geometrie