Math 647 FALL 2006, List of lectures

  • On this page I will post content of all lectures with reference to the book. All handouts also will be posted here.
  • Monday, September 25: Preview: ruler and compass constructions.
  • Wednesday, September 27: Extensions of fields; finite and algebraic extensions; structure of simple extensions; finitely generated algebraic extension is finite. Homework is available!
  • Friday, September 29: Splitting field of a polynomial; count of homomorphisms from a splitting field; any two splitting fields for a given polynomial are isomorphic.
  • Monday, October 2: Multiple roots of irreducible polynomials. Perfect fields.
  • Wednesday, October 4: Galois extensions; Galois group and its order.
  • Friday, October 6: The fundamental theorem of Galois theory. New Homework!
  • Monday, October 9: Gauss's Lemma and Eisenstein's criterion. Examples for the fundamental theorem.
  • Wednesday, October 11: Roots of unity of prime degree.
  • Friday, October 13: Roots of unity of arbitrary degree. New Homework!
  • Monday, October 16: Cubics and quartics. Discriminant.
  • Wednesday, October 18: Normal extension of normal extension is not always normal. A polynomial with Galois group S_5 over Q.
  • Friday, October 20: Solvable groups. Equation solvable in radicals has solvable Galois group. New Homework!
  • Monday, October 23: Equation with solvable Galois group is solvable in radicals.
  • Wednesday, October 25: Algorithm for computing Galois group. Reductions modulo prime.
  • Friday, October 27: Algebraic closure: existence and uniqueness. There is no homework due next Friday but there will be MIDTERM soon!!!
  • Monday, October 30: The field of complex numbers is algebraically closed. Theorem on primitive element.
  • Wednesday, November 1: Normal basis Theorem. We will have MIDTERM on Tuesday, November 7, 10-12am; we meet near Deady 210.
  • Friday, November 3: Group actions. The class equation and consequences. See Rotman's book Section 2.7. New Homework!
  • Monday, November 6: Finite abelian groups. Rotman, Section 5.1.
  • Tuesday, November 7: Midterm; here are solutions.
  • Wednesday, November 8: The Sylow Theorems. Rotman, Section 5.2.
  • Friday, November 10: Smallest nonabelian simple group. Sylow p-subgroup of GL(n,q). Rotman, Section 5.2. New Homework!
  • Monday, November 13: Alternating groups are simple. Projective unimodular groups. Rotman, Sections 2.7, 5.4.
  • Wednesday, November 15: PSL(2,q) are simple for q>3. Applications. Rotman, Section 5.4.
  • Friday, November 17: The Jordan-Holder Theorem. Rotman, Section 5.3. New Homework!
  • Monday, November 20: More on solvable groups. Nilpotent groups. Rotman, Section 5.3.
  • Monday, November 27: Free group. Rotman, Section 5.5.
  • Wednesday, November 29: Presentations by generators and relations; von Dyck's Theorem. Rotman, Section 5.5.
  • Friday, December 1: The Nielsen-Schreier Theorem. Rotman, Section 5.6.
  • THE END