# Math 541

Notes on Math 541: Modern Algebra @ University of Wisconsin-Madison

Homepage

Syllabus

# Textbook

Abstract Algebra, by Dummit and Foote, Third Edition, 2004

# Lecture Notes

Week 1
1/24 Divides, Equivalence Relations
1/26 Well-ordering of Z
Week 2
1/29 Division Algorithm, gcd
1/31 Euclidean Algorithm
2/2 Equivalence Class, Z/nZ, Group
Week 3
2/5 Group, Well-definedness, Z/nZ
2/7 (Z/nZ)*, Properties of Group
2/9 Order, Symmetric Group
Week 4
2/12 Symmetric Group, Cycle
2/14 Homomorphism, Isomorphism
2/16 Order, Homomorphism, Subgroup
Week 5
2/19 Dihedral Groups, Subgroup
2/21 Cyclic Group, lcm, Order of g^a
2/23 Cyclic Subgroup, Generating Set of a Group
Week 6
2/26 Finitely Generated Group
2/28 Coset, Normal Subgroup
3/2 Exam 1
Week 7
3/5 Quotient Group, Index, Lagrange's Theorem
3/7 Corollaries of Lagrange's Theorem
3/9 The First & Second Isomorphism Theorems
Week 8
3/12 The Third Isomorphism Theorem
3/14 Transposition, Sign of Permutation
3/16 Homework 6, The Correspondence Theorem
Week 9
3/19 Sign of Permutation, Alternating Group
3/21 Subgroups of A_4, Group Action, Orbit, Stabilizer
3/23 Orbit, Stabilizer, Cayley's Theorem
Week 10
4/2 Conjugacy Class, The Class Equation
4/4 Cauchy's Theorem, Recognizing Direct Products
4/6 Homework 8, Properties of Finite Abelian Group
Week 11
4/9 Fundamental Theorem of Finite Abelian Groups
4/11 Definition of Ring
4/13 Exam 2
Week 12
4/16 Properties of Ring, Zero-Divisor, Unit
4/18 Field, Product Ring, Integral Domain
4/20 Product Ring, Finite Domain and Field, Subring
Week 13
4/23 Polynomial Ring, Ideal, Principal Ideal
4/25 Examples of Ideals, Quotient Ring
4/27 Isomorphism Theorems for Rings
Week 14
4/30 Generating Ideal, Maximal Ideal, Prime Ideal
5/2 Prime Ideal, Euclidean Domain
5/4 Review, Galois Theory