Modern Algebra I
Course number: 21770
Meeting MWF 11:00-11:50
San Diego State University
Final Exam: Mon. Dec. 15, 10:30-12:30
Professor: Mike O'Sullivan
Web page: http://www.rohan.sdsu.edu/~mosulliv
Office: GMCS #579, ext. 594-6697
Office Hours: MWF 8-9, MW 1-2,
You may also make an appointment for another time or stop by my
If I am in and available, we can talk.
Notes on group theory.
The ultimate goal of this course is to introduce the main theorems and
standard examples in Galois theory. Along the way we will cover
the fundamentals of groups, commutative rings, and fields.
The roots of Galois theory lead back to
problems posed by the ancient Greeks and their predecessors.
Greek geometers achieved remarkable constructions with ruler and
compass, but a number of simple, nagging, problems remained
unresolved until the Rennaisance. For example: Is it possible to
trisect an arbitrary angle? Which regular polygons are constructible?
In algebra, several civilizations investigated the solution of a
quadratic equation (see
MathWorld article ). The attempt to find a solution for higher
degree equations was another project that occupied numerous
The resolution of these ancient questions culminated in Galois' theory
of fields. It is a delightful subject, and the modern treatment
highlights the interplay between three key areas of algebra:
groups, rings and fields.
Abstract Algebra: The basic graduate year .
This book is a concise and direct treatment of the fundamentals of
graduate level algebra.
Hungerford, Abstract Algebra: An Introduction 2nd ed.
This has been the standard text for undergraduate algebra at SDSU for
a few years. It will be useful for review of material that is covered
tersely in Ash's book.
A good understanding
of the basics of groups, rings and fields (Math 521A and 521B is
I will assume you are conversant with the following material, and
need only a gentle reminder. I suggest you review the main points in
the sections from Hungerford noted below.
- The integers (H 1.1-3, 2.1-2): The division theorem, greatest
common divisor, Euclidean algorithm, prime numbers, unique
factorization, modular arithmetic.
- Polynomial rings in one variable over a field (e.g. the
rationals) (H. 4.1-4, 5.1-2):
The division theorem, greatest
common divisor, Euclidean algorithm, irreducible polynomials, the
correspondence between linear factors of a polynomial and roots, unique
factorization, congruence modulo a polynomial.
- Commutative rings (H 3.1-3, 6.1): We will only treat commutative rings, and they
will usually be derived from the integers or a polynomial ring over
the rationals. You should be familiar with the basic definitions:
ideals, homomorphisms, integral domain.
- Linear Algebra: Vector space over a field, nullspace, subspace,
dimension, linear independence, spanning set, basis.
- Groups (H. 7.1-5): Definitions of group, subgroup, homomorphism.
The most basic results (e.g. uniqueness of the inverse).
Standard examples like Z_n (integers mod n), D_n (the dihedral group),
S_n (the symmetric group). Cosets, index, Lagrange's theorem.
Some of the following topics will be familiar from your
undergraduate course. We will cover them in greater depth, and with
more attention to details. I expect this material to take 6-9 weeks.
- Groups: Normal subgroups and the isomorphism theorems. Groups
acting on sets, orbits and stabilizers. The Sylow theorems.
- Commutative Rings: Ideals, homomorphisms and quotient rings.
The isomorphims theorems. Prime and maximal ideals. Criteria for
irreducibility of polynomials.
- Fields: Field extensions. Adjoining a root of a polynomial.
Splitting field of a polynomial.
The main topics are
- Separable, inseparable, and normal field extensions.
- The Galois group of a field extension.
- The fundamental theorem: relating intermediate fields of K/F
to subgroups of the Galois group of K/F
- Standard examples: finite fields, cyclotomic fields, solvability
by radicals, constructable numbers.
There will be several (6-8) homework assignments a midterm and a final exam.
The final grade will be weighted as follows.