Math 649 SPRING 2023, List of lectures

  • On this page I will post content of all lectures with reference to the book.
  • Monday, April 3: ED, PID, UFD. Section 9.1.
  • Wednesday, April 5: Gauss' Lemma etc. Section 9.2
  • Friday, April 7: Field extensions. Section 10.1.
  • Monday, April 10: More on field extensions. Galois group. Sections 10.2, 11.1.
  • Wednesday, April 12: Field of invariants under finite group action. Section 11.1.
  • Friday, April 14: Splitting fields and normal extensions. Section 11.2.
  • Monday, April 17: Normal extensions and separable extensions. Sections 11.3-11.4.
  • Wednesday, April 19: Fundamental Theorem of Galois Theory. Section 11.5.
  • Friday, April 21: Proof of the Fundamental Theorem of Galois Theory. Galois group of a polynomial. Sections 11.5, 11.6.
  • Monday, April 24: Discriminant. Radical extensions. Sections 11.6, 12.4.
  • Wednesday, April 26: Radical extensions are solvable. Section 12.4.
  • Friday, April 28: Kummer's theory. Primitive element theorem. Normal basis theorem. Algebraic closure. Finite fields and cyclotomic fields. Sections 12.4, 13.1-13.4.
  • Monday, May 1: Transcendence bases. Section 13.5.
  • Wednesday, May 3: Properties of transcendence degree. Symmetric functions. Sections 13.5-13.6.
  • Friday, May 5: Prime ideals, multiplicative sets, localization. Sections 19.1-19.2.
  • Monday, May 8: More on localization. Section 19.2.
  • Wednesday, May 10: Integral extensions. Section 19.3.
  • Friday, May 12: Midterm.
  • Monday, May 15: Prime ideals in integral extensions. Section 19.4.
  • Wednesday, May 17: Noether Normalization Lemma. Section 19.4.
  • Friday, May 19: Hilbert basis theorem. Section 20.1.
  • Monday, May 22: Primary ideals. Section 20.2.
  • Wednesday, May 24: Primary decomposition of ideals and associated primes. Section 20.2.
  • Friday, May 26: Isolated and embedded primes. Algebraic geometry. Sections 20.2-21.1.
  • Monday, May 29: Memorial Day, no classes
  • Wednesday, May 31: Nullstellensatz. Section 21.1.
  • Friday, June 2: Zariski topology. Section 21.2.
  • Monday, June 5: Regular functions and regular maps. Category of algebraic sets is equivalent to (opposite) category of affine algebras. Sections 21.3-21.4.
  • Wednesday, June 7: Products of algebraic sets. Rational functions. Sections 21.5-21.6.
  • Friday, June 9: Sheaf of regular functions. Dimension. Sections 21.6-21.7.