Math 371 Algebra
Semester: Spring 2007
Lecture 

Office:   DRL 4N40 
Phone:   (215) 8988460 
Office hours:   TBA or by appointment


Text book:   Michael Artin, Algebra, Prentice
Hall: 1991. 

Lab 
T.A.:  
Asher Auel
auela AT math.upenn.edu 
Time:   Lab 101 Tue 6:308:30 pm
Lab 102 Thu 6:308:30 pm 
Loct:   DRL 4C2 

Office:   DRL 3E2 
Phone:   (215) 8988175 
Office hours:   Tue 3:00  4:00 pm Wed
5:30  6:30 pm or by appointment


This course is a continuation of Math 370
from Fall 06. 

Homework: The currently assigned homework and occasionally
posted solutions to problems can be found at the homework page.
Syllabus: This course will touch on two major theories
combining various topics we've covered so far:
 Representation theory: combines group theory and linear
algebra to gain a better understanding of how groups (we'll be mostly
concentrating on finite groups) arise as "groups of matrices." We'll cover:
 A brief review of linear algebra: linear maps, vector space bases,
change of bases, dual spaces, and bilinear forms.
 The theory of semisimple algebras: group rings and endomorphism
rings.
 Introduction to group representations, and the dictionary between
representations and "modules for the group ring," irreducible
representations, and Shur's Lemma.
 Character theory: class functions, Ginvariant pairings,
characters of representations, orthogonality relations, character
tables, and many examples.
 Operations on representations: tensor product, reduced
representations, inflation, induced representations, and Frobenius
reciprocity.
 Applications: to pure groups theory and the modern theory of
particle physics.
 Galois theory: combines groups theory and field theory to
better understand roots of polynomials equations. If we have time,
we'll cover:
 Algebraic extensions of fields: root fields and splitting
fields.
 Galois groups: field automorphisms, entension of automorphisms,
normal extensions, the Galois group, the main theorem of Galois
theory.
 Applications: solvable groups and the insolvability of the general
quintic equation and classical problems in geometry.
Policies
(or otherwise the small print)
Homework: I will be grading your homework and tests for this
course. For general policies regarding homework, tests, grade
breakdown, homework lateness, illnesses, etc., please see
Prof. Gerstenhaber. Homework will be assigned during Tuesday's
lecture and will be due the following Tuesday, by 4 pm, in my mailbox
in the mathematics department main office. Special arrangements can
be made ahead of time if you're physically unable to hand in your
homework (like emailing it to me if you're trapped on a desert island
with internet access.)
Generally, a homework problem in this course (and in general any
mathematics problem) will consist of two parts: the creative
part and the writeup.
 The creative part: This is when you "solve" the problem.
You stare at it, poke at it, and work on it until you understand
what's being asked, and then try different ideas until you find
something that works. This part is fun to do with your friends, and
during this part, if you're having trouble, even in understanding what
the problem's asking, you should come ask Prof. Gerstenhaber or myself
for hints, either in person during office hours, or by email. This
part should all be done on "scratch paper."
 The writeup: Now that everything about the problem is
clear in your mind, you go off by yourself and write up a coherent,
succinct, wellwritten, and grammatically correct mathematical
solution on clean sheets of paper. Consider this your final
draft, just as in any other course. This part you should
definitely NOT do with your friends.
This will most likely be your first "real" math course, in the sense
that some problems will require an argument and not just a long
calculation. In order to write an argument, you'll have to use words
to convey your ideas and how they connect together. Yes, it may seem
strange that in a math course you'll have to use the English language.
As in any other course where you use the English language, you'll need
to use it correctly, i.e. you must use complete sentences, correct
grammar and spelling, etc. It's true that mixing correct English with
mathematical symbols is somewhat of an art, but the author of your
text, for example, provides a good example of how to do this
successfully. Also, if your handwriting is illegible, then consider
typing up your papers, as you would for your English class.
Please note that a fully correct solution requires both parts: having
"figured out" the problem, but not having written it up (or having
written up something incoherent that does not express what you know)
or conversely, having written up many pages of beautiful prose that
still fail to solve the problem, don't count for very much. You will
be graded accordingly.
Links:   
Mathematics Department links
Math Help Resources

