Lectures for Mathematics 332, Spring 2009-10
- Monday, 1/25: Overview, begin the integers
- Tuesday, 1/26: More on the integers
- Wednesday, 1/27: Finish handout on the integers
- Friday, 1/29: Start preview of groups by
examples
- Monday, 2/1: More preview of groups
- Tuesday, 2/2: Definition of group;
basic properties;
subgroup, subgroup tests
- Wednesday, 2/3: More on subgroups; homomorphisms
- Friday, 2/5: [Add/section-change deadline]
Group actions: centralizer, normalizer as isotropy
- Monday, 2/8: Cyclic Groups
- Tuesday, 2/9: Kernels and quotients
- Wednesday, 2/10:
Cosets in Lagrange's Theorem and in actions
- Friday, 2/12: The symmetric group,
simplicity of the alternating group
- Monday, 2/15: Multiplicativity of indices, Cayley's Theorem,
first isomorphism theorem
- Tuesday, 2/16: Isomorphism theorems, solvable groups
- Wednesday, 2/17: Solvable groups,
abelian quotients,
- Friday, 2/19: Group products,
p-adic numbers as limit
- Monday, 2/22: Finitely-generated abelian groups
- Tuesday, 2/23: The Sylow theorems
- Wednesday, 2/24: Finish Sylow theorems, start
classification of small groups
- Friday, 2/26: More classification of small groups
- Monday, 3/1: [No-W drop deadline]
Finish classification of small groups
- Tuesday, 3/2: Loose ends
- Wednesday, 3/3: Start Euclidean rings
- Friday, 3/5: Finish Euclidean rings
- Monday, 3/8: Ideals of a commutative ring,
quotient field of an integral domain
- Tuesday, 3/9: Algebraic structure and generative
derivative rules
- Wednesday, 3/10: What is a polynomial?
- Friday, 3/12: Informal meeting
- Spring break week
- Monday, 3/22: The p-adic integers
classically and as a limit
- Tuesday, 3/23: Factorization of polynomials
- Wednesday, 3/24: Fundamental theorem of symmetric polynomials, discriminant
- Friday, 3/26: Solving cubic and quartic polynomials via resolvents
- Monday, 3/29: Resultants
- Tuesday, 3/30: Cyclotomic polynomials: definition, examples,
integrality, co-primality
- Wednesday, 3/31: Cyclotomic polynomials: Infinitely many p=1modn,
Wedderburn's Theorem, irreducibility, reducibility mod p
- Friday, 4/2: Finish cyclotomic polynomials,
start free modules
- Monday, 4/5: [Withdraw/leave deadline]
Finish free modules
- Tuesday, 4/6: Start modules over a PID
- Wednesday, 4/7: More modules over a PID
- Friday, 4/9: Finish modules over a PID
- Monday, 4/12: Canonical forms in linear algebra,
start tensor products
- Tuesday, 4/13: Tensor product: definition, uniqueness, construction
- Wednesday, 4/14: Tensor product: multiplicativity of free rank
under tensor product, preservation of rank under extension of
scalars, tensor product of maps
- Friday, 4/16: Tensor product formation as left adjoint of restriction
- Monday, 4/19: Exterior products: uniqueness, existence, generators
- Tuesday, 4/20: Exterior products: rank, mappings, determinants
- Wednesday, 4/21: Damage repair, completing a matrix
- Friday, 4/22: Uniqueness revisited for modules over a PID
- Monday, 4/26: Intrinsic proof of
the Cayley-Hamilton Theorem
- Tuesday, 4/27: The seventeenth root of
unity via quadratics
- Wednesday, 4/28: Last discussion of Cayley-Hamilton, evaluations
- Friday, 4/30: (No meeting)
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