(* See also: *
[ vignettes ]
...
[ functional analysis ]
...
[ intro to modular forms ]
...
[ representation theory ]
...
[ Lie theory, symmetric spaces ]
...
[ buildings notes ]
...
[ number theory ]
...
[ algebra ]
...
[ complex analysis ]
...
[ real analysis ]
...
[ homological algebra ]
)

...
[* 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.

[ 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, Gaussian and Eisenstein integers, 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
- hmwk/examples 01
[
*updated*Thu, 07 Sep '23, 03:01 PM] ... discussion 01 [*updated*Mon, 15 Jan '24, 05:26 PM]

- hmwk/examples 01
[
- 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
- hmwk/examples
02
[
*updated*Tue, 03 Oct '23, 04:36 PM] ... discussion 02 [*updated*Tue, 17 Oct '23, 04:17 PM]

- hmwk/examples
02
[
- 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
- hmwk/examples 03
[
*updated*Tue, 14 Nov '23, 02:07 PM] ... discussion 03 [*updated*Tue, 14 Nov '23, 03:15 PM]

- hmwk/examples 03
[
- 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
- hmwk/examples 04
[
*updated*Sat, 18 Nov '23, 12:54 PM] ... discussion 04 [*updated*Sat, 18 Nov '23, 12:51 PM]

- hmwk/examples 04
[
- 12 polynomials over UFDs: Gauss' lemma, fields of fractions, examples
- 13 symmetric groups: cycles,
disjoint cycle decomposition, transpositions, examples
- alternating groups, commutators
- small Wedderburn theorem : finite rings without zero divisors are (commutative!) fields
- hmwk/examples 05
[
*updated*Mon, 15 Jan '24, 05:10 PM] ... discussion 05 [*updated*Mon, 15 Jan '24, 05:11 PM]

- 14 Naive set theory
- 15 Symmetric polynomials
- 16 Eisenstein's criterion for irreducibility: prime-order cyclotomic polynomials and other iconic examples
- 17 Vandermonde determinants (again!)
- 18 Cyclotomic polynomials II
- 105th cyclotomic polynomial
- prototypes of quadratic reciprocity, via cyclotomic fields
- hmwk/examples 06
[
*updated*Thu, 15 Feb '24, 04:50 PM] ... discussion 06 [*updated*Thu, 15 Feb '24, 04:51 PM]

- 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
- hmwk/examples 07
[
*updated*Sat, 17 Feb '24, 05:07 PM] ... discussion 07 [*updated*Sat, 17 Feb '24, 12:06 PM] - hmwk/examples 07b
[
*updated*Tue, 09 Apr '24, 04:16 PM] ... discussion 07b [*updated*Tue, 09 Apr '24, 04:16 PM]

- hmwk/examples 07
[
- 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
- hmwk/examples 08 (toooo many?!?)
[
*updated*Tue, 09 Apr '24, 05:01 PM] ... discussion 08 [*updated*Tue, 09 Apr '24, 04:51 PM]

- hmwk/examples 08 (toooo many?!?)
[

Elementary exercises and notes: [Intro to Abstract Algebra]

Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |

Jan 17 | Jan 19 | |||||

Jan 22 | Jan 24 | Jan 25 | ||||

Jan 29 | Jan 31 | Feb 02 | ||||

Feb 05 hmwk 05 | Feb 07 | Feb 09 exam 05 | ||||

Feb 12 | Feb 14 | Feb 16 | ||||

Feb 19 | Feb 21 | Feb 23 | ||||

Feb 26 hmwk 06 | Feb 28 | Mar 01 exam 06 | ||||

Mar 04 | Spring | Mar 06 | Break | Mar 08 | ||

Mar 11 | Mar 13 | Mar 15 | ||||

Mar 18 | Mar 20 | Mar 22 | ||||

Mar 25 | Mar 27 | Mar 31 | ||||

Apr 01 hmwk 07 | Apr 03 | Apr 05 exam 07 | ||||

Apr 08 | Apr 10 | Apr 12 | ||||

Apr 15 | Apr 17 | Apr 19 | ||||

Apr 22 hmwk 08 | Apr 24 | Apr 26 exam 08 | ||||

Apr 29 last class |

Unless explicitly noted otherwise, everything here, work by Paul Garrett, is licensed under a Creative Commons Attribution 3.0 Unported License. ... [

The University of Minnesota explicitly requires that I state that