Abstract Algebra
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[ garrett@math.umn.edu ]
The main prerequisite for 8201 is good
understanding of undergrad algebra and linear algebra, with substantial experience writing
proofs .
Students coming into this course should have a range of
experience in proof writing, not only in a previous course
in abstract algebra, but also in analysis, rigorous linear
algebra, and some point-set topology. All these play significant
roles, both directly, and in terms of mathematical maturity and
vocabulary.
Coherent writing is essential. Contrary to some myths,
the symbols do not speak for themselves.
Prerequisite for 8202: 8201 or equivalent.
Grades fall and spring will be determined
by four in-class midterms , scheduled as below. You are
not competing against other students in the course, and I
will not post grade
distributions. Rather, the grade regimes are roughly
90-100 = A, 75-90 = B, 65-75 = C, etc., with finer gradations of
pluses and minuses. So it is possible that everyone gets a "A", or
oppositely. That is, there are concrete goals, determined by what
essentially all mathematicians need to know, and would be happy
to know.
There will be optional homework/example assignments preparatory to
exams, as scheduled below, on which I'll give feedback about
mathematical content and writing style. The homeworks will not
directly contribute to the course grade, and in principle are
optional, but it would probably be unwise not to do them
and get feedback. No late homeworks will be accepted. Homework should
be typeset, presumably via (La)TeX, and submitted by
email. The notes contain discussions/solutions of the
homework/examples. If you find useful things in prior years' example
discussions, or elsewhere on the internet, or in books, cite
. Also, collaboration with other people is fine, and
acknowledge . It is ok to learn from other people, I
think. :)
This course is not a gauntlet to be run. The course is about
increasing awareness and exposure to
important, useful (also crazy and entertaining) ideas, so that in the
future when they show up (seemingly out of the blue?) in your work,
you can recognize them and act accordingly.
Text is below, with a few future updates along the way.
In 2023-24, MWF 11:15-12:05, Vincent 301, office hours after
class MW, email anytime
-->
[ My book/notes on abstract algebra ] ... (updated
Sat, 21 Jul '07, 12:39 PM) ... in individual chapters
below. Various additions will be made along the way, but these notes
are 90% correct as to what we'll cover.
Miscellaneous notes:
Solutions to standard exercises:
s01 ,
s02 ,
s03 ,
s04 ,
s05 ,
s06 ,
s07 ,
s08 ,
s09 ,
s10 ,
s11 ,
s12 ,
s13 ,
s14 ,
s15 ,
s15b ,
s16 ,
s17 ,
s18 ,
s19 ,
s20 ,
s21
Course notes
... individual chapters from notes linked-to above:
- 01 the integers: unique
factorization, integers mod m, Fermat's little theorem, Sun-Ze's
theorem, examples.
- 02 groups I: subgroups, Lagrange's
theorem, homomorphisms, kernels, normal subgroups, cyclic groups,
quotient groups, groups acting on sets, Sylow theorems, worked examples
- 03 the players: rings, fields, etc:
homomorphisms, vectorspaces, modules, algebras, polynomial rings I
- 04 commutative rings I: divisibility
and ideals, polynomials in one variable over a field, ideals and
quotients, maximal ideals and fields, prime ideals and domains,
Fermat-Euler on sums of two squares, examples
- 05 linear algebra I: dimension,
bases, homomorphisms
- 06 fields I: adjoining things,
fields of fractions, fields of rational functions, characteristics,
finite fields, algebraic field extensions, algebraic closures
- 07 some irreducible polynomials:
over a finite field, examples
- 08 cyclotomic polynomials: multiple
factors in polynomials, finite subgroups of fields, infinitude of
primes p=1 mod n, examples
- 09 finite fields: uniqueness,
Frobenius automorphism, counting irreducibles
- 10 modules over PIDs: structure
theorem, variations, finitely-generated abelian groups, Jordan
canonical form, conjugacy versus k[x]-module isomorphism, examples
- 11 finitely-generated modules: free
modules, finitely-generated modules over a domain, PIDs are UFDs,
structure theorem (again), submodules of free modules
- 12 polynomials over UFDs: Gauss'
lemma, fields of fractions, examples
- 13 symmetric groups: cycles,
disjoint cycle decomposition, transpositions, examples
- 14 naive set theory: sets, posets,
ordinals, transfinite induction, finiteness/infiniteness, comparison
of infinities, transfinite Lagrange replacement, equivalents of the Axiom of Choice
- 15 symmetric polynomials:
discriminants, examples
- 16 Eisenstein's criterion: examples
- 17 Vandermonde determinants: examples
- 18 cyclotomic polynomials II: over
the integers, examples
- 19 roots of unity: cyclotomic
fields, solutions in radicals, Lagrange resolvents, quadratic
fields, quadratic reciprocity, examples
- 20 cyclotomy III: prime power
cyclotomic polynomials over the rationals, irreducibility,
factoring, examples
- 21 primes in arithmetic
progressions: Euler and the zeta function, Dirichlet's theorem, dual
groups of abelian groups, non-vanishing on the line Re(s)=1,
analytic continuations, Dirichlet series with positive coefficients
- 22 Galois theory: field extensions,
imbeddings, automorphisms, separable field extensions, primitive
elements, normal extensions, Galois' theorem, conjugates, trace,
norm, examples
- 23 solution by radicals: Galois'
criterion, composition series, Jordan-Holder theorem, solving cubics
by radicals, examples
- 24 eigenvectors, eigenvalues,
spectral theorems: diagonalizability, semi-simplicity, commuting
endomorphisms ST=TS, inner product spaces, projections without
coordinates, unitary operators, spectral theorems, corollaries, examples
- 25 duality, naturality, bilinear
forms: dual vectorspaces, examples of naturality, bilinear forms, examples
- 26 determinants I: prehistory,
definitions, uniqueness, existence
- 27 tensor products: desiderata,
uniqueness, existence, tensor products of maps, extension of
scalars, functoriality, naturality, examples
- 28 exterior algebra: desiderata,
uniqueness, existence, exterior powers of free modules, determinants
revisited, minors of matrices, uniqueness in the structure theorem,
Cartan's lemma, Cayley-Hamilton theorem, examples
Elementary exercises and notes:
[Intro to Abstract Algebra]
| Sunday |
Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
Saturday |
| | |
Sept 06 | | Sept
08 | |
| Sept 11 | | Sept
13 | | Sept 15 | |
| Sept 18 | | Sept
20 | | Sept 22 | |
| Sept 25 hmwk 01 | | Sept
27 | | Sept 29 exam 01 | |
| Oct 02 | | Oct
04 | | Oct 06 | |
| Oct 09 | | Oct 11 |
| Oct 13 | |
| Oct 16 hmwk 02 | | Oct
18 | | Oct 20 exam 02 | |
| Oct 23 | | Oct
25 | | Oct 27 | |
| Oct 30 | | Nov
01 | | Nov 03 | |
| | Nov 06 | | Nov
08 | | Nov 10 | |
| Nov 13 hmwk 3 | | Nov
15 | | Nov 17 exam 03 | |
| Nov 20 | | Nov
22 | Thanksgiving |
Nov 24 | |
| Nov 27 | | Nov 29 | | Dec 01 | |
| Dec 04 hmwk 04 | | Dec 06
| | Dec 08 exam 04 | |
| Dec 11 | | Dec 13 last
class | | | |
Unless explicitly noted otherwise, everything here, work
by Paul Garrett, is licensed
under a Creative
Commons Attribution 3.0
Unported License.
...
[ garrett@umn.edu ]
The University of Minnesota explicitly requires that I
state that "The views and opinions expressed in this page are
strictly those of the page author. The contents of this page have not
been reviewed or approved by the University of Minnesota."