School of Mathematics
School of Mathematics
Course 311 - Abstract Algebra 2007-08 (Optional JS & SS Mathematics, SS Two-subject Moderatorship
Lecturer: Dr. D.R. Wilkins
Duration: 18 weeks
Number of lectures per week: 3
End-of-year Examination: One 3-hour examination
See http://www.maths.tcd.ie/~dwilkins/Courses/311/ for
lecture notes and more detailed information.
- Chapter 1: Topics in Group Theory.
This chapter begins with a brief review of basic group theory.
This is followed by the statement and proof of the
Sylow Theorems. These theorems are then applied in order
to prove that all simple groups of order less than 60 are cyclic.
The chapter concludes with a discussion of solvability, a concept
that is of key importance in Galois Theory.
- Chapter 2: Rings and Polynomials.
This chapter begins with a brief review of the definitions and
basic properties of rings, integral domains and fields. This is
followed by a detailed discussion of the basic properties of
rings of polynomials in one variable with coefficients in a field.
These results are very important for the development of Galois
Theory. We prove Gauss's Lemma, which ensures that a polynomial
whose coefficients are all integers can be factored as a product
of polynomials of lower degree with rational coefficients if and
only it can be factored as a product of polynomials of lower degree
with integer coefficients. Another important result concerning
irreducibility discussed here is Eisenstein's Irreducibility
- Chapter 3: Introduction to Galois Theory.
This chapter concerns the application of concepts and results from
group theory, ring theory and field theory to the study of polynomial
equations. Here one seeks to express the roots of a polynomial as functions
of its coefficients. To any polynomial is associated a finite group,
referred to as the Galois group of the polynomial. The roots of a
polynomial can be expressed in terms of its coefficients by means of
algebraic formulae involving only the operations of addition,
subtraction, multiplication, division and the extraction of nth roots
if and only if the Galois group of the polynomial is `solvable'. This
result can be used to prove that there cannot exist any algebraic formula
for the roots of a general quintic polynomial that involves only the
algebraic operations of addition, subtraction, multiplication, division
and the extraction of nth roots.
- Chapter 4: Modules and Commutative Rings.
This chapter covers some of the basic concepts and results of
commutative algebra. We introduce the concept of a
module over a commutative ring. We study Noetherian
rings and modules: a Noetherian ring is a unital commutative
ring in which every ideal is finitely generated: a Noetherian
module is a module over a unital commutative ring in which every
submodule is finitely generated. We shall prove Hilbert's Basis
Theorem, which ensures that any ring of polynomials with coefficients
in a Noetherian ring is itself a Noetherian ring. The chapter concludes
with a proof of a classification theorem for finitely generated modules
over a principal ideal domain.
- Chapter 5: Algebraic Varieties and Hilbert's Nullstellensatz.
This chapter provides an introduction to basic concepts of algebraic
geometry, which is concerned with the study of sets of common
zeros of collections of polynomials in several indeterminates.
Any collection of polynomials in n indeterminates (or variables)
with coefficients in a field K determines a corresponding subset
of Kn (the set of all ordered n-tuples of elements of K).
This subset is the set of common zeros of the polynomials
in the collection, and sets of this form are referred to as
algebraic sets. We show that there is a well-defined topology
on the set Kn, referred to as the Zariski topology, whose
closed sets are the algebraic sets in Kn. We also examine the
correspondence between algebraic sets in Kn and ideals of the
corresponding polynomial ring. The deepest theorem in this section
of the course is Hilbert's Nullstellensatz.
The Weak Nullstellensatz is essentially a generalization
of the Fundamental Theorem of Algebra. It asserts that the set
of common zeros of a collection of polynomials in n indeterminates
with coefficients in an algebraically closed field K is non-empty
if and only if that collection generates a proper ideal of the
corresponding polynomial ring. The Strong Nullstellensatz
establishes a one-to-one correspondence between algebraic sets
and radical ideals of the polynomial ring, in the case where the
field of coefficients is algebraically closed.
Oct 3, 2007
File translated from
On 3 Oct 2007, 16:43.