George M. Bergman -- graduate course materials
These are handouts I have refined over the years in teaching
Berkeley's first graduate algebra course, in the order in which
I generally use them.
They can be viewed or downloaded as PostScript files;
if you can't do this I can send hardcopy.
(The source files are in locally enhanced troff, so I can't
send TEX files.)
Generally used in 250A (groups, rings, fields, Galois theory):
-
The Axiom of Choice, Zorn's Lemma, and all that.
Proves the equivalence of AC, Zorn and well-orderability, and that
these imply comparability of cardinals (Bernstein's Theorem), with
sketch of converse.
Aimed at students in the abovementioned
course, who have seen the definition of partially ordered set.
4pp., last revised Spring '97.
-
A principal ideal domain that is not Euclidean, developed as a
series of exercises.
Proof that Z[(1+(-19)½)/2]
has the asserted property.
1p., last revised Fall '95.
-
Luroth's Theorem and some related results,
developed as a series of exercises.
Proof that a field between k and
k(t) has the form k(u).
2pp., last revised Spring '97.
-
Solution in radicals of polynomials of degree _<4,
developed as a series of exercises.
Assumes only standard facts about symmetric polynomials, and
the arithmetic of primitive cube roots and 4th roots of unity.
2pp., last revised Fall '95.
-
Quadratic reciprocity,
developed from the theory of finite fields as a series of
exercises.
Assumes Galois theory, finite fields, basic facts about the
discriminant of a polynomial.
2pp., last revised Spring '97.
Generally used in 250B (``additional topics''):
-
Infinite Galois theory, Stone spaces, and profinite
groups.
Automorphism groups of infinite Galois extensions as inverse limits.
Examples.
Profinite topology.
Correspondence between subfields and closed subgroups.
All topological groups with Stone topology are profinite, but
not so for lattices etc..
Assumes acquaintance with inverse limits, p-adic integers,
point-set topology.
12pp., last revised Spring '97.
-
Notes on composition of maps.
It is often useful to write morphisms of left modules to the
right of their arguments, and morphisms of right modules to the
left, and compose each sort accordingly. This note (emphasizing
but not limited to module theory) discusses advantages,
drawbacks, and consequences of such notational conventions.
6pp., last revised Spring '97.
-
Tensor algebras, exterior algebras and symmetric algebras.
Tensor products and exterior and symmetric powers of modules;
their universal properties; the graded algebra structure on their
direct sum; relations with determinants; further observations.
10pp., last revised October 2011.
And here are my course-notes for Math 245 (also linked to my
main web page):
The remaining three items are not course handouts, but I
include them because they are also relevant to graduate study.
-
On mock quals, and a condition I require of
students who ask me to serve on their Qualifying Exam committees.
(1 p.)
-
Advice on the language examinations, based
on having graded many of them.
(2 pp.)
-
Answers to student's questions.
In my upper division and graduate courses, I require every student
to submit a question on each day's reading.
I incorporate the answers to some of these into my lectures;
others I answer by e-mail.
Recently, I have begun maintaining cumulative web-pages of answers
sent by e-mail on points that seemed important enough that other
students in the class might want to refer to them.
Here are the files for
Math 250A,
Fall 2002 and Fall 2006, taught
from Langs's Algebra.
Math 245,
Spring 2008, Fall 2011, Spring 2014, Fall 2015, and Fall 2017,
taught from the abovementioned
course notes,.
In each of these,
some of the answers have been edited to improve clarity etc.
Unfortunately the answers to the questions asked by large numbers of
students may not be in this file, since they are answered in class.
The files of the same sort for my undergraduate courses
can be found on my page of
undergraduate course materials