lecture 1 (08/26/24) Review of well-ordering, induction. Divisibility of integers. Existence and uniqueness of gcd's. Notion of a prime. Factorization into primes. There are infinitely many primes.
lecture 2 (08/28/24) Equivalence relations and partitions. Integers mod n form a ring. Definition of a ring.
lecture 3 (08/30/24) Definition of a group and of a homomorphism. Basic properties of homomorphisms.
lecture 4 (09/04/24) Isomorphisms, subgroups, criterion for being a subgroup, kernel and image of a homomorphism, subgroup generated by an element, a homomorphism is injective iff the kernel is trivial.
lecture 5 (09/06/24) Group actions. Orbits. Orbits of an action partition a set. Right cosets of a subgroup.
lecture 6 (09/09/24) Left cosets of a subgroup. A homomorphism descends to an injective map from the set of cosets of the kernel. Classification of subgroups of Z. Any finite cyclic group is isomorphic to Zn .
lecture 7 (09/11/24) A subgroup generated by a subset. The dihedral group Dn by generators and relations.
lecture 8 (09/13/24) Permutation groups. Cycles. Disjoint cycles commute. Any permutation is a product of disjoint cycles.
lecture 9 (09/16/24) Uniqueness of decomposition of a permutation into a product of disjoint cycles. Orbit/stablizer theorem. All cosets are the same size.
lecture 10 (09/18/24) Finite orbits for an action of Z are cyclic. Lagrange's theorem. Left and right cosets are in general different. Normal subgroups. For a normal subgroup left and right cosets are the same.
lecture 11 (09/20/24) Kernels are normal subgroups. Quotient by a normal subgroup is a group. First isomorphism theorem.
lecture 12 (09/23/24) 2nd and 3d isomorphism theorems. Product of two groups.
lecture 13 (09/25/24) Product of two groups. Sign representation of the symmetric group Sn .
lecture 14 (09/27/24) Conjugacy classes, centralizers, the class equation, classification of groups of order p2 (p a prime).
lecture 15 (09/29/24) Cauchy's theorem. Semi-direct products.
lecture 16 (10/02/24) Semi-direct products. Statement of Syllow theorems.
lecture 17 (10/04/24) Statement of Sylow theorems. Groups of order pq (p,q primes). The group of automorphisms of Zn.
review for the first midterm (10/07/24)
first midterm exam (10/09/24)
lecture 18 (10/11/24) Rings. Basic properties. Polynomial rings.
lecture 19 (10/14/24) Ring homomorphisms. Subrings. Universal property of polynomial rings. Evaluation maps and "change of scalars". Ideals. Kernels of homomorhisms are ideals.
lecture 20 (10/16/24) Quotient rings. First isomorphism theorem for rings.
lecture 21 (10/18/24) Ideals generated by a subset and principal ideals. Not all ideals are principal. Zero divisors. Integral domains. Finite integral domains are fields. If D is an integral domain, then for any polynomials f,g in D[x], deg(fg) = deg(f) + deg(g).
lecture 22 (10/21/24) Division algorithm for F[x] and principal ideals. Roots of polynomials. F[x]/(p).
lecture 23 (10/23/24) Product of rings. Maximal ideals. M is maximal in a comm. ring R iff R/M is a field. Prime ideals. P is prime in a comm ring R iff R/P is an integral domain. Maximal ideals are prime.
lecture 24 (10/25/24) Irreducibles. Notion of PID. Euclidean domains. Gaussian integers. A Euclidean domain is a PID. Primes in commutative rings. In integral domains nonzero primes are irreducibles but not conversely.
lecture 25 (10/28/24) In a PID irreducibles are primes and give rise to maximal ideals. Associates. Ascending chains of ideals. Union of an ascending chain is an ideal. In a PID ascending chains terminates. Existence of factorization into irreducibles in a PID.
lecture 26 (10/30/24) PIDs are unique factorization domains (UFDs). Products of ideals. Definition of a module.
lecture 27 (11/01/24) Submodules. F[x]-module structures on a vector space over F and linear maps. Homomorphisms, kernels and images of homomorphisms. Quotient modules. First isomorphism theorem for modules.
review for the second midterm (11/04/24)
second midterm exam (11/06/24)
lecture 28 (11/08/24) Generators. Cyclic modules. Linear independence. Basis. Direct sums of modules.
lecture 29 (11/11/24) External and internal direct sums of modules. Statement of the structure theorem for finitely generated modules over a PID (2 versions).
lecture 30 (11/13/24) Chinese remainder theorem and its connection with the structure theorem(s) for modules over PIDs. Annihilators and torsion in a module.
lecture 31 (11/15/24) Chinese remainder theorem. Definition of torsion.
lecture 32 (11/18/24) Torsion submodules. Any finitely generated module is a quotient of a free module. Surjective map to a free module splits.
lecture 33 (11/20/24) Dual basis of Hom(M, R) for a finitely generated free R-module M. Start of proof that if M' is a submodule of a free finitely generated module M over a PID R, then M has a basis adapted to M'.
lecture 34 (11/22/24) The rest of the proof of the existence of adapted basis. Application to finitely generated modules over a PID -- existence of a normal form.
lecture 35 (12/02/24) End of proof of existence part of the structure theorem for modules over PID. Start of discussion of Jordan normal form.
lecture 36 (12/04/24) Proof of Jordan normal form theorem. Classification of finite abelian groups. Start of proof of the uniqueness part of the structure theorem for modules over PIDs.
lecture 37 (12/06/24). Uniqueness part of the structure theorem for finitely generated modules over a PID.
lecture 38 (12/09/24) Last bits.
review (12/11/24)
Last modified: Th November 21 15:20:36 CST 2024