lecture 1 (01/22/25) 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 (01/24/25) Equivalence relations and partitions. Integers mod n form a ring. Definition of a ring.
(01/27/25) lecture cancelled
lecture 3 (01/29/25) Definition of a group and of a homomorphism. Basic properties of homomorphisms.
lecture 4 (01/31/25) 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 (02/03/25) Group actions. Orbits. Orbits of an action partition a set. Right cosets of a subgroup.
lecture 6 (02/05/25) 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 group generated by one element is isomorphic to Zn . The orthogonal and special orthogonal groups. Intersection of a family of subgroups is a subgroup.
lecture 7 (02/07/25) A subgroup generated by a subset. The dihedral group Dn by generators and relations.
lecture 8 (02/10/25) Permutation groups. Cycles. Disjoint cycles commute. Any permutation is a product of disjoint cycles.
lecture 9 (02/12/25) Uniqueness of decomposition of a permutation into a product of disjoint cycles. Orbit/stablizer theorem. Finite orbits for an action of Z are cyclic. All cosets (left and right) are the same size. Lagrange's theorem. Index of a subgroup.
lecture 10 (02/14/25) Left and right cosets are in general different. Normal subgroups. For a normal subgroup left and right cosets are the same. Kernels are normal subgroups. Cosets of a normal subgroup form a group.
lecture 11 (02/17/25) Order of an element. In finite groups g|G| = e for all g in G. First isomorphism theorem. Third isomorphism theorem.
lecture 12 (02/19/25) 2nd isomorphism theorem. Any permutation is a product of transpositions. The representation of the symmetric group by symmetric matrices.
lecture 13 (02/21/25) Sign representation of the symmetric group Sn. Centralizers, conjugacy classes .
lecture 14 (02/24/25) The class equation, classification of groups of order p2 (p a prime), Cauchy's theorem: if a prime p divides the order of a group G then G has an element of order p.
lecture 15 (02/26/25) Semi-direct products. Automorphisms of Zn .
lecture 16 (02/28/25) Automorphisms of Zn and an application. Normalizers, conjugate subgroups. Statement of Syllow theorems.
lecture 17 (03/03/25) Groups of order pq (p,q primes). Review for the first midterm.
first midterm exam (03/05/25)
lecture 18 (03/07/25) Proofs of Sylow theorems.
lecture 19 (03/10/25) Rings. Basic properties. Polynomial rings.
lecture 20 (03/12/25) Ring homomorphisms. Subrings. Universal property of polynomial rings. Evaluation maps and "change of scalars". Ideals. Kernels of homomorhisms are ideals.
lecture 21 (03/14/25) Quotient rings. First isomorphism theorem for rings. Intersection of a family of ideals is an ideal. Principal ideals.
lecture 22 (03/24/25) 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 23 (03/26/25) Division algorithm for F[x] and principal ideals. Roots of polynomials. F[x]/(p).
lecture 24 (03/26/25) Product of rings. Maximal ideals. M is maximal in a comm. ring R iff R/M is a field. Prime ideals. P is a prime ideal in a comm ring R iff R/P is an integral domain. Maximal ideals are prime.
lecture 25 (03/31/25) Irreducibles. Notion of PID. Euclidean domains. Gaussian integers. A Euclidean domain is a PID. Primes in commutative rings. In integral domains primes are irreducibles but not conversely.
lecture 26 (04/02/25) 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.
review for the second midterm (04/04/25) Problem 10 corrected.
second midterm exam (04/07/25)
lecture 27 (04/09/25) PIDs are unique factorization domains (UFDs). Products of ideals. Definition of a module and some properties of modules. Submodules. F[x]-module structures on a vector space over F and linear maps.
lecture 28 (04/11/25) 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. Generators. Cyclic modules.
lecture 29 (04/14/25) Linear independence. Basis. Direct sums of modules (external and internal). Statement of the structure theorem for finitely generated modules over a PID [invariant factors form].
lecture 30 (04/16/25) Chinese remainder theorem and its connection with the structure theorem(s) for modules over PIDs. Annihilators and torsion in a module.
lecture 31 (04/18/25) Definition of torsion. Torsion submodules. Any finitely generated module is a quotient of a free module. Surjective map to a free module splits.
lecture 32 (04/21/25) Rank of a finitely generated free module. Existence of maximal ideals (no proof). If M is a free module of rank m over a PID and N is a submodule then N is also free and rank N is at most m.
lecture 33 (04/23/25) Dual basis of Hom(M, R) for a finitely generated free R-module M. If N is a submodule of a free finitely generated module M over a PID R, then M has a basis adapted to N.
lecture 34 (04/25/25) The rest of the proof of the existence of adapted basis and an example.
lecture 35 (04/28/25) Structure theorem for finitely generated modules over a PID in various forms. Proof of the existence part. Classification of finite abelian groups.
lecture 36 (04/30/25) Start of proof of the uniqueness part of the structure theorem for finitely generated modules over a PID.
lecture 37 (05/02/25) End of proof of uniqueness part. Start of Jordan normal form discussion.
lecture 38 (05/05/25) Proof of Jordan normal form theorem
review (05/07/25)
Last modified: Saturday, May 3 15:20:36 CST 2025