## Modern Algebra: Math 627B, Spring 2012 Schedule

### Schedule

My best, but rough, approximation.
[A] Ash, Abstract Algebra: The basic graduate year.
[H] Hungerford, Abstract Algebra: An Introduction 2nd ed.
[OS] O'Sullivan, Lecture Notes for Math 627, Modern Algebra.
[JB] Judson, Beezer Abstract Algebra: Theory and Applications Available free online.

Free groups.
Generators and relations
Day Topics Preparation
Th. 1/19 Introduction: Groups, Rings, Fields.
Groups, subgroups and homomorphisms.
[OS] Sec. 1.
Tu. 1/26 Order theorem. Automorphism group.
Lattice of subgroups, Symmetric Group.
[OS] Sec. 1-2
Th. 1/26 Groups in sage.
Meet in lab GMCS 425
[JB] Ch 3-6 Sage exercises.
Tu. 1/31 Alternating group
Cosets, Index of a group,Cayley's theorem.
Conjugation.
[OS] Sec. 2-3
Th. 2/2 Rings and unit groups. [OS].
Tu. 2/7 Irreducibility in polynomial rings.
Quotients of F[x] .
[OS].
Th. 2/9 Cubic field extensions. [OS].
Tu. 2/14 Finite Fields. [OS].
Th. 2/16 Automorphisms of Finite Fields: The Frobenius map. [OS].
Tu. 2/21 Irreducible over the rationals, Q
[OS]
Th 2/23 Roots of a cubic equation
and field extensions.
[OS]
Tu 2/28 Normal subgroups and quotient groups
The factor theorem.
[A] Sec. 1.2-3
Th 3/1 The 1st and 3rd isomorphism theorems and the correspondence theorem
Matrix groups examples.
[A] Sec. 1.4
Tu 3/6 The second isomorphism theorem. Direct and semidirect products. [A] Sec. 1.4-5
Th 3/8 Direct and semidirect products. [A] Sec. 1.4-5
[A] Sec. 5.8
Tu. 3/13 Groups acting on a set. [A] 5.1-2.
Th. 3/15 Orbit/stabilizer theorem.
Groups of prime-power order
First Sylow theorem.
[A] 5.1-2.
Tu.3/20 The second and third Sylow theorem.
Application of Sylow theorems.
[A] 5.4-5.
Th. 3/22 Abelian groups. [H] 8.2 (or other sources).
Tu. 4/3 Algebraic field extensions [A] 3.1-2; [H] 10.4.
Th. 4/5 Some group theory problems.
Algebraic extensions and splitting fields.
[A] 3.1-2; [H] 10.4.
Tu. 4/10 TEST: Groups. .
Th. 4/12 Splitting fields and algebraic closure.
Separable extensions.
[A] 3.3-5
Tu 4/17 Primitive element Theorem.
Normal and Galois extensions.
The Galois correspondence.
[A] 6.1-2; [H] 12.2.
Th 4/19 The fundamental theorem of Galois theory. [A] 6.2; [H] 12.2.
Th 4/24 Cyclotomic fields extensions. [A] 6.3,6.
[A] 6.5
Th 4/26 Cyclic and abelian extensions. [A] 6.7.
Tu 5/1 Geoemtric constructions. [A] 6.8.
Th. 5/3 Commutator subgroups and solvable groups. [A] 5.7.
Tu. 5/8 Solvability by radicals [A] 6.8