Welcome to Math 200A!

Course description:

This is the first course in the three-part 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.you-know-where.edu, AP&M 6-101.

Lectures: MWF, 11pm-11:50pm, AP&M 5-402.

Office hours:

Wednesday 3-4 in AP&M 6-101.

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 12-1, Wednesday 4-5, or by appointment. Office: APM 6-351.

Textbook: D. Dummit, R. M. Foote, Abstract Algebra, 3rd edition.

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, 11:30AM-2:30PM in 5-402.

Announcements

• Wednesday, Sept 29 only: we meet in 6-402 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 q-1.
• Lecture 5: More on automorphisms of groups. Groups of order p^2q when p does not divide q-1. 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 q-1. 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 p-group 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 Hasse-Dedekind 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 Hasse-Dedekind 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