Modern Algebra: Math 627A, Fall 2012

Day Topics Reading
Tu. 8/28 Introduction: Groups, Rings, Fields.
Groups, subgroups and homomorphisms.
[OS] Sec. 1.
[A] Sec 1.1
Th. 8/30 Order theorem.
Lattice of subgroups.
[OS] Sec. 1-2
[A] Sec. 1.2
Tu. 9/4 The symmetric Group.
Groups in sage.
Meet in lab GMCS 422
[JB] Ch 3-6 Sage exercises.
Th. 9/6 Cartesian product of groups.
Automorphism groups. The alternating group
[OS] Sec. 1-2
[A] Sec. 1.3
Tu. 9/11 Cosets, index of a group,Cayley's theorem.
[OS] Sec. 3
Th. 9/13 Rings and unit groups. [OS] Sec. 4-5.
[H] Sec. 3.1-2
Tu. 9/18 The main theorems for Polynomial Rings. [OS] Sec. 8-10.
[H] Ch 4,5.
Th. 9/20 Polynomial ring modulo a polynomial. [OS] Sec. 8-10.
[H] Ch 4,5.
Tu. 9/25 More on polynomial rings
Mason Stothers theorem and the abc conjecture.
[OS] Sec 10.
Th. 9/27 Irreducibility in polynomial rings.
Quotients of F[x] .
[OS] Sec. 10-11
[H] Sec. 4.5.
Tu. 10/2 Fields, homomorphisms, and automorphism groups.
The quadratic formula and quadratic extensions of Q
[OS] Sec 12-13.
Th. 10/4 The cubic formula and cubic extensions of Q. [OS] Sec. 13-14.
Tu. 10/9 Finite Fields. [OS] Sec. 15.
Th 10/11 Finite Field structure.
Automorphisms and containment
[OS] Sec. 15.
Tu 10/16 Normal subgroups and quotient groups The factor theorem.
The 1st and 3rd isomorphism theorems and the correspondence theorem.
[A] Sec. 1.4
Th 10/18 The second isomorphism theorem. Direct and semidirect products. [A] Sec. 1.4-5
Tu 10/23 Direct and semidirect products.
Exact sequences and classification of groups.
[A] Sec. 1.4-5
Th. 10/25 Abelian groups.
Free groups. Generators and relations
[H] 8.2 (or other sources)[A] Sec. 5.8
Tu. 10/30 Problem solving party. .
Th. 11/1 TEST .
Tu. 11/6 Groups acting on sets
Orbit/stabilizer theorem.
Groups of prime-power order.
[A] 5.1-2.
Th. 11/8 Proof and applications of the Sylow theorems. [A] 5.4-5.
Tu. 11/13 Algebraic field extensions.
Splitting fields and algebraic closure.
[A] 3.1-2; [H] 10.4.
Th. 11/15 Separable extensions.
Primitive element Theorem.
[A] 3.3-5
Tu. 11/20 Normal and Galois extensions.
The Galois correspondence.
[A]3.5, 6.1-2; [H] 12.2.
Tu. 11/27 The fundamental theorem of Galois theory.
Cyclotomic fields extensions.
[A] 6.2; [H] 12.2.
[A] 6.5
Th 11/29 Cyclic and abelian extensions. [A] 6.7.
Tu 12/4 Geometric constructions. [A] 6.8.
Th 12/6 Commutator subgroups and solvable groups.
Solvability by radicals
[A] 5.7.


[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.