Modern Algebra: Math 627B, Spring 2016


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 627B, Modern Algebra.
[JB] Judson, Beezer Abstract Algebra: Theory and Applications Available free online.

Day Topics Preparation
Th. 1/21 Introduction: Some classical problems.
Constructible numbers, solution of x^3-2 .
[OS] Sec. 17, 18, 19.
Tu. 1/26 Fundamental properties of polynomial rings over a field.
Cubic field extensions of Q .
[OS] Sec. 10-15, 19.
Th. 1/28 Derivatives and irreducibility.
Mason Stothers theorem and the abc conjecture.
[A] Sec. 2.9 [OS] Sec. 16.
Tu. 2/2 Mason Stothers theorem and the abc conjecture.
Lattice of subgroups. Properties of the direct product.
[OS] Secs 5.
Tu. 2/4 Matrix groups. Automorphism groups. [OS] Sec 10.
Tu. 2/9 Symmetric and Alternating group. Simplicity of A_n
[OS] Sec. 2
Th. 2/11 The second isomorphism theorem.
Direct and semidirect products.
Group Extensions
[A] Sec. 1.4-5
Tu. 2/16 Classification of abelian groups. [H] Sec. 8.2
Th. 2/18 Free groups and presentations.
Cyclic extensions.
[A] Sec 5.8.
Tu. 2/23 No class: do problems .
Th. 2/25 TEST: Groups. .
Tu. 3/1 Orbit-stabilizer theorem [A] 5.1,2
Th. 3/3 Online videos: Sylow Theorems [A] 5.1-4.
Tu. 3/8 Several problems on group actions. Groups of prime power order. .
Th. 3/10 Applications of Sylow theorems.
A theorem about semi-direct products.
Groups of order pq^r
[A] 5.4,5.
Tu. 3/15 Groups of order 27.
Composition series for p -groups .
[A] 5.5 and
Groups of Small Order .
Th. 3/17 Jordan-Holder theorem: uniqueness of factors in the composition series.
Solvability and nilpotency. .
Tu. 3/22 More on solvability, Sylow and group actions. .
Th. 3/24 TEST: Group Actions, Sylow theorems, group presentations, semi-direct products. .
Tu. 4/5 Finite Fields. [OS] Sec 20. [A] Sec 6.4.
Th. 4/7 Automorphisms of Finite Fields: The Frobenius map. [OS] Sec 20 [A] Sec 6.4.
Tu. 4/12 Separable and nonseparable extensions.
The primitive element theorem.
Field homomorphism.
[A] Secs 3.1-4. [OS]. 22-25.
Th. 4/14 Galois correspondence. [OS] Sec 27. [A] 6.1-2.
Tu. 4/19 Galois main theorem. [OS] Sec 27. [A] 6.1-2.
Th. 4/21 Cyclotomic fields extensions. [A] 6.3,6.
[A] 6.5
Tu. 4/26 Cyclic and abelian extensions. [A] 6.7.
Tu 4/28 Geometric constructions. [A] 6.8.
Tu. 5/3 Solvability by radicals [A] 6.8
Th. 5/5
Tu. 5/10 Final Exam .