Dirk Schütz's Algebra page

Literature

We do not follow a book strictly, but the material can be found in standard textbooks, some of which are listed below.

RBJT Allenby, Rings, fields and groups : an introduction to abstract algebra, Arnold, 1991.
PJ Cameron, Introduction to Algebra, Oxford University Press, 1998.

For the group theory part, an introduction is given by

MA Armstrong, Groups and Symmetry. Springer, 1988.

Cameron also has the following very nice online Notes on Algebraic Structures which cover similar topics and are in some respects more detailed.

Handouts

Here is a file with notes on what has been covered so far. Please note that examples are not very detailed and proofs are omitted so you should still get your notes from the lectures. Also, there may be several typos in them and I would appreciate if you can point them out to me.

Problem Class	Date	Solutions
pdf	17.10.	pdf
pdf	31.10.	pdf
pdf	14.11.	pdf
pdf	28.11.	pdf
pdf	12.12.	pdf

Tutorial	Week	Solutions
pdf	3	pdf
pdf	5	pdf
pdf	7	pdf
pdf	9	pdf

Lecture Outline

Date	Outline
10.12.	In this lecture we define groups and subgroups, and see some basic examples.
05.12.	In this lecture we study maximal ideals and use them to construct fields out of irreducible polynomials.
28.11.	In this lecture we prove the Chinese Remainder Theorem, see some applications of it, and define prime ideals.
26.11.	In this lecture we will see some examples of quotient rings and prove the first Isomorphism Theorem.
21.11.	In this lecture we begin our study of quotient rings.
19.11.	In this lecture we show Fermat's Little Theorem and analyze which prime numbers can be written as the sum of two squares.
14.11.	In this lecture we show that Euclidean Domains are PIDs, show that the Gaussian Integers form a Euclidean Domain, and look at Unique Factorization Domains.
12.11.	In this lecture we define an ideal, see some examples, and introduce the notion of a Principal Ideal Domain.
07.11.	In this lecture we use the Gauss Lemma to give criteria for irreducibility of polynomials over the rationals, including the Eisenstein criterion.
05.11.	In this lecture we show that polynomials over a field factorise uniquely into irreducibles, and consider irreducible polynomials over the complex, real and rational numbers.
29.10.	In this lecture we deal with irreducible and prime elements, in particular for polynomials.
24.10.	In this lecture we explain the Euclidean algorithm for polynomials, show some examples and look at roots of polynomials.
22.10.	In this lecture we start with a more systematic analysis of the ring of polynomials over a field.
17.10.	In this lecture we prove Bezout's identity, define fields and deal with divisors.
15.10.	In this lecture we define kernel and image of ring homomorphisms, integral domains and units.
10.10.	In this lecture we look at subrings, the division algorithm and ring homomorphisms.
08.10.	In this lecture we define rings, see some examples and basic properties.

Algebra II