1/22: Revision and introduction to modules
1/24: Examples of modules and submodules
1/29: Quotient modules and homomorphism
1/31: School closed!
2/5: The first isomorphism theorem, the forth isomorphism theorem, and external direct sum
2/7: Internal direct sum and the submodule Rx
2/12: Structure of cyclic modules, Introduction to free modules, and the universal property of free modules
2/14: The rank of free modules, matrix algebra, and change of basis matrices
2/19: Basis of submodules, row and column operations, Smith normal form, and relation matrix
2/21: Decomposition of a finitely generated module over a PID (an existence part)
2/26: Decomposition of a finitely generated module over a PID (a uniqueness part)
2/28: Primary decomposition theorem and elementary divisors
3/5: Introduction to F[l]-module associated to a linear operator on a finite dimensional vector space
2/7: Rational canonical form and minimal polynomials
3/12: Jordan canonical form
3/14: Revision
Here is my lecture note before midterm 1
Honors homework about tensor products of modules in Dummit & Foote
Disclaimer: there are possibly some errors in my homework!
Midterm 2015 + Solution (Thanks Prof. Tullia for an old midterm)
3/26: Definitions of subfield and extension, the minimal polynomial
3/28: Characterization of algebraic elements and multiplicity of degree
4/2: The extension of a field by several generators and introduction to finite fields
4/4: Existence and Uniqueness of F(u)
4/9: Automorphism of fields and Extension Theorem for Simple Extension
4/11: The proof of Extension Theorem for Simple Extension and introduction to Galois groups
4/16: Spitting fields, Extension Theorem for Splitting Fields
4/18: Normal extension, Galois groups over normal extension, and finite fields
4/23: Classification of finite fields
4/25: Invariant group and Artin's lemma
4/30: Characterization of Normal Extensions
5/2: Galois Correspondence and Fundamental Theorem of Galois Theory
5/4: The proof of Fundamental Theorem of Galois Theory
Here is my lecture note before final
Honors homework about solvable and radical extensions in Dummit & Foote