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Modern Algebra: Math 627A, Fall 2008
Schedule


Schedule

My best, but rough, approximation.
[A] Ash, Abstract Algebra: The basic graduate year
[H] Hungerford, Abstract Algebra: An Introduction 2nd ed.

Splitting fields and algebraic closure.
Day Topics Preparation
Wed. 9/3 Introduction: Groups, Rings, Fields.
Galois Theory.		 .
Fri. 9/5 Solution of the cubic equation. .
Mon. 9/8 Groups and homomorphism, subgroups. [A] Sec. 1.1, 1.3
Wed. 9/10 Order, Lagrange's theorem, cosets, conjugacy. [A] Sec. 1.3
Fri. 9/12 Normal subsgroups. Quotient groups. [A] Sec. 2.1-2
Mon. 9/15 The 1st and 3rd isomorphism theorems. [A] Sec. 1.4
Wed. 9/17 Abelian groups. [H] Sec. 8.2
Fri. 9/19 Abelian groups. [H] Sec. 8.2
Mon. 9/22 The 2nd isomorphism theorem, semidirect and direct products. [A] Sec. 1.4
Wed. 9/24 Permutation groups. [A] Sec. 1.2 [H] Sec. 7.9,10
Fri. 9/26 Permutation groups. [A] Sec. 1.2 [H] Sec. 7.9,10
Wed. 10/1 Matrix groups.
The quaternions.		 .
Fri. 10/3 Free groups, defining groups by
generators and relations. 		[A] 5.8.
Mon. 10/6 Groups acting on a set. [A] 5.1-2.
Wed. 10/8 Groups acting on a set.
Orbits and stabilizers. 		[A] 5.1-2.
Fri. 10/10 Solutions to problem set II. [A] 5.1-2.
Mon. 10/13 More solutions..
Orbits/ stabilizer theorem. 		[A] 5.1-2.
Wed. 10/15 Class equation.
Classification of groups of order 8. 		[A] 5.2.
Fri. 10/17 The Sylow theorems.
Two proofs of the first theorem.
Groups of order pq . 		[A] 5.4-5.
Mon. 10/20 Proofs of other Sylow theorems. [A] 5.4-5.
Wed. 10/22 Commutator subgroups and solvable groups. [A] 5.7.
Fri. 10/24 Simple groups.
The simplicity of A_n . 		[A] 5.6, Problems 4-9.
Mon. 10/27 Subnormal and normal series. The refinement theorem and the Jordan-Holder theorem. [A] 5.6.
Wed. 10/29 Solutions to 3rd assignment. .
Fri. 10/31 Questions before exam. .
Mon. 11/3 EXAM. Ch 5. problems listed on HW4.
Wed. 11/5 Fields and extensions.
Sec. 3.1
Fri. 11/7 Algebraic extensions. {A] 3.1; [H] Ch. 4.
Mon. 11/10 Polynomial ring over a field: main thms.
Roots and Factors. 		[A] 3.1; [H] Ch. 4.
Wed. 11/12 Polynomial ring modulo a polynomial. [A] 3.1; [H] Ch. 5.
Fri. 11/14 [A] 3.2-3; [H] 10.4.
Mon. 11/17 Finite Fields. [A] 6.4; [H] 10.6.
Wed. 11/19 Irreduciblity over Q .
Spitting field of x^3-2 . 		[A] 6.3.
Fri. 11/21 Homomorphsisms of fields. [A] 3.2.3, 3.2.5, 3.5.1, 3.5.2
A.3.1, A.3.5.
Mon. 11/24 Homomorphsisms of fields. [A] 3.2.3, 3.2.5, 3.5.1, 3.5.2
A.3.1, A.3.5.
Wed. 11/26 Separable extensions. [A] 3.4; [H] 10.5
Mon. 12/1 Normal extensions. [A] 3.4
Wed. 12/3 The Galois correspondence. [A] 6.1,2; [H] 12.2.
Fri. 12/5 The fundamental theorem of Galois theory. [A] 6.2; [H] 12.2.
Mon. 12/7 Cyclotomic fields. [A] 6.5.
Wed. 12/9 The Galois group of x^2-4 . [A] 6.3,6.
Fri. 12/11 Solutions of cubics and quartics. [A] 6.6.