Math 200A  Graduate Algebra
Welcome to Math 200A!
Course description:

This
is the first course in the threepart graduate algebra sequence. The first
quarter will cover groups and rings.
The course will
assume that you've already had reasonable exposure to groups, rings, and
fields at undergraduate level.
Instructor: Dragos Oprea, doprea "at" math.youknowwhere.edu,
AP&M 6101.
Lectures: MWF, 11pm11:50pm, AP&M 5402.
Office hours:
Wednesday 34 in AP&M 6101.

I
am
available for questions after lecture or by appointment. Also, feel
free
to drop in if you see me in my office.
Teaching Assistant:
Joel Dodge, jrdodge "at"
you know
where.

Joel's office hours: Tuesday 121, Wednesday 45, or by
appointment. Office: APM 6351.
Textbook: D. Dummit, R. M. Foote, Abstract Algebra, 3rd
edition.
Grading:
Homework (30%), Midterm (30%), Final
(40%) or Homework (10%), Midterm (30%), Final (60%).
Exams:
 The midterm is in class, on November 10. The
final exam will be in class, Tuesday, December 7, 11:30AM2:30PM in 5402.
Announcements
 Wednesday,
Sept 29 only: we meet in 6402 at 1pm.
Lecture Summaries
 Lecture
1: Groups, examples, cyclic groups. Group homomorphisms, isomorphisms.
Subgroups, centralizer, normalizer.
 Lecture 2: Lattice of subgroups. Quotient groups and
normal subgroups. Lagrange's theorem. Isomorphism theorems. Simple groups.
Composition series.
 Lecture 3: Group actions, stabilizer. More on group actions:
(i) groups acting on themselves by left multiplication, Cayley's theorem
(ii) groups acting on their quotients, applications.
 Lecture 4: (iii) Action of a group on itself by conjugation,
the class equation. Groups of order p^2. (iv) Action by conjugation on a
normal subgroup. Groups of order pq when p does not divide q1.
 Lecture 5: More on automorphisms of groups. Groups of order
p^2q when p does not divide q1. Cauchy's theorem. Groups of order 2p.
Sylow's theorems stated and
some examples.
 Lecture 6: Proof of Sylow's theorems.
 Lecture 7: More applications of Sylow: remarks on groups of
order 12, 30, 60. Simplicity of A_5.
 Lecture 8: Simplicity of A_n. Direct products
defined. Recognizing direct products.
 Lecture 9: Semidirect products. Recognizing
semidirect products.
 Lecture 10: Examples of semidirect products. Groups with pq
elements classified when p divides q1. Groups with 30 elements
classified. Examples of groups with p^3 elements.
 Lecture 11: Groups with p^3 elements classified.
Groups with 12 elements classified.
 Lecture 12: Groups with 8 elements classified. Structure
theorem for finite abelian groups.
 Lecture 13: Commutators. Derived series. Solvable
groups.
 Lecture 14: Nilpotent groups. Every pgroup is nilpotent.
Every nilpotent group is solvable. Stated the classification of nilpotent
groups.
 Lecture 15: Rings, domains, fields, ideals. Examples.
Isomorphism theorems for rings. Operations on ideals: sum, product.
 Lecture 16: Maximal ideals. Existence of maximal ideals. Every
ideal is contained in a maximal ideal. Prime ideals and examples. Radical
ideals, nilradical. Nilradical is the intersection of prime ideals.
 Lecture 17: Multiplicatively closed subsets; two main
examples. Rings of fractions. Localization at prime ideals.
 Lecture 18: Geometric interpretation of A_f and A_p for prime
ideals p when A is a polynomial ring. A_p is a local ring. Prime ideals in
S^{1}A.
 Lecture 19: Coprime ideals and the Chinese remainder theorem.
Special classes of rings (begun): Euclidean domains and examples.
 Lecture 20: Principal ideal domains. Greatest common divisor.
Euclidean domains are PID's and the gcd can be found by the Euclidean
algorithm. Quadratic integers for D=5 do not form a PID, quadratic
integers for D=19 do not form an Euclidean domain.
 Lecture 21: Principal ideals domains and HasseDedekind norms.
Example: quadratic
integers in Q(sqrt(19)) form a PID. Prime and irreducible elements.
Prime implies irreducible. Irreducible implies prime in PID's.
 Lecture 22: Unique factorization domains. Principal ideals
domains are UFDs. Existence of HasseDedekind norms in PID. The primes in
Z[i] and sums of two squares.
 Lecture 23: Polynomial rings: k[x] is Euclidean. Gauss' lemma,
A[x] is UFD if A is UFD.
 Lecture 24: Applications for polynomials over
fields: number of roots bounded by degree, finite subgroups of
multiplicative groups in fields are
cyclic, units in F_p form a cyclic group. Applications of
polynomials over UFDs: irreducibility criteria,
Eisenstein crieterion, irreducibility of cyclotomic polynomials.
 Lecture 25: Noetherian rings. Examples and first properties.
The nilradical is nilpotent. Hilbert basis theorem. Artin rings. Examples
and first propreties (quotients, rings of fractions). Prime ideals in
Artin rings are maximal.
 Lecture 26: Artin rings have finitely many maximal ideals. The
Jacobson radical is nilpotent. Examples of local Artin rings. Any Artin
ring is
product of local Artin rings.
 Lecture 27: Arting rings are Noetherian. Krull dimension.
Artin rings have zero Krull dimension, but not conversely, unless we
assume Noetherian. Krull dimension interpreted topologically. Krull
dimension 1: DVRs. Definition and examples.
 Lecture 28: Structure theorem for DVRs. Dedekind domains.
Localizations of Dedekind domains are DVRs. Examples. Fractional ideals.
Ideal class group.
 Lecture 29: In a Dedekind domain, every fractional ideal is
invertible. Every fractional ideal is product of prime ideals in a unique
way. Cool diagram establishing connections between various
types of rings we studied.
Homework
 Homework 1: PDF due Oct 1.
Problem 17 page 35 was meant to be Problem 16 page 33.
 Homework 2: PDF due Oct 8.
 Homework 3: PDF due Oct 15.
 Homework 4: PDF due Oct 22.
 Homework 5: PDF due Oct 29.
 Homework 6: PDF due Nov 5.
 Homework 7: PDF due
Nov 17.
 Midterm: PDF and Solutions.
 Homework 8: PDF due
Dec 3. Note the changed due date.
 Final: PDF and Solutions.