Course Description
Fall Semester 2005
Math 4281 (Introduction to Modern Algebra)
MWF 9:05 a.m. to 9:55 a.m. VinH 6
Instructor: Prof. Joel Roberts
Posted on Wednesday, January 11, 2006
Some general information.
This is an
introductory course in modern algebra. It differs from
Math 5285H: Fundamental Structures of Algebra by (i) being less
theoretical, and (ii) having somewhat different subject matter.
(There are, however, some overlaps in topics covered in the two courses.)
The only prerequisite for Math 4281, aside from sophomore level calculus-type
courses is Math 2283 or Math 3283W (Sequences, Series, and
Foundations) -- either version is acceptable. (I'll explain below
what this prerequisite actually has to do with the present course.)
The text for Math 4281 is
A concrete introduction to higher algebra, by Lindsay Childs.
We'll cover the equivalent of about one average sized chapter (or a respectable
fraction of it) each week. Some sections will be skipped, especially in the
longer chapters, and some entire chapters will be skipped. It's not that
I consider those sections or chapters to be unimportant or uninteresting.
(In fact, quite the opposite is true in some cases.) Instead, it's that we
have only a limited number of weeks, and there are quite a few basic
concepts of algebra which we really need to cover.
Main topics to be covered will
include:
- Greatest common divisor (GCD) of two integers (Chapters 3 and 4)
- Finding it without factoring
(Euclid's algorithm)
- Applications of Euclid's algorithm (to some math questions)
- Congruences (Chapters 5 and 6)
- This is related to the elementary topic of
modular arithmetic,
but there's ¡quite a bit more!
that we can do in this direction.
- Rings and fields (Chapter 8)
- Algebraic structures where addition and multiplication are possible
- Theorems of Fermat and Euler (Chapter 9)
- Powers of an element in modular arithmetic
- Groups (Chapter 11)
- An algebraic system with just 1 operation (usually called
multiplication),
but with far-reaching consequences
- The Chinese Remainder Theorem (Chapter 11)
- Solutions of a system of congruences, not all to
the same modulus
- Matrices, etc. (part of Chapter 13)
- Polynomials and factorization (Chapters 14 & 15; parts of
16 & 18)
- Many ideas about factoring integers, including Euclid's algorithm,
can be adapted to factoring polynomials in 1 variable.
Our orientation toward proofs.
As you can see from our list of topics, there's a lot that
we can learn about algebra even without a huge emphasis on proofs. And indeed,
we probably won't insist on mastering all of the details of complicated proofs.
On the other hand, there are some facts ¡about integers
already! that can't be effectively understood without use of the method
of mathematical induction. This is the main thing that we'll need from
Math 2283/3283, in addition to basic ideas of what a proof is all about.
In other words, some of the "Foundations" part of Math 2283/3283 is what's
relevant to Math 4281; the "Sequences and Series" material has
little or no applicability here. We don't expect you to start
the course with a high level of proficiency in doing proofs by induction, but
everyone needs to develop at least a basic working understanding of that method.
For information about
required work and expectations,
please see the course syllabus.
Info relating to the math major requirements:
Math 4281 is
on the Algebra List for the
Upper Division
Math Requirements, actually being listed in the Column X portion of the
list.
For further information: Please
send me an e-mail, or call me
at the phone number listed below.
Back to the class homepage.
Prof. Joel Roberts
School of Mathematics
University of Minnesota
Minneapolis, MN 55455
USA
Office: 351 Vincent Hall
Phone: (612) 625-1076
Dept. FAX: (612) 626-2017
e-mail:
roberts@math.umn.edu
http://www.math.umn.edu/~roberts