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

[Old Course Notes] ... (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:

- Poincare-Birkhoff-Witt theorem
- Girard-Newton formulas
- small yoneda lemma, adjoint functors
- semi-simple algebras

** 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]

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