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.
Wednesday, October 4: Galois extensions; Galois group and its order.
Friday, October 6: The fundamental theorem of Galois theory.
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.
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
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.
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.