Math 366: Introduction to Abstract Algebra
Colorado State University, Spring 2021
Instructor: Henry Adams
Email: henry dot adams at colostate dot edu
Lectures: MWF 12:00-12:50pm Mountain Time on Zoom
Textbook: Abstract Algebra: Theory and Applications by Thomas W Judson. This book is freely available as a PDF, and you can find more resources at the book webpage. The book has selected exercise hints in the back, which is great!
Overview: This course is a rigorous and proof-based introduction to abstract algebra. Topics covered include sets, integers, polynomials, real and complex numbers, groups, integral domains, and fields.
Syllabus: Here is a course syllabus.
Course notes: Here are Henry's course notes from teaching a prior version of this class in Spring 2020 and 2019.
You do not need to do your homework assigments in LaTeX; I include the LaTeX source files for your homework below only if it is convenient for you, and in the spirit of making things publicly available!Homework 1 (LaTeX Source) is due Monday, February 1.
Homework 2 (LaTeX source) is due Friday, February 19.
Homework 3 (LaTeX source) is due Friday, March 12.
Homework 4 (LaTeX source) is due Friday, March 26.
Homework 5 (LaTeX source) is due Friday, April 9.
Homework 6 (LaTeX source) is due Friday, April 30.
Homework 7 (LaTeX source) is due Wednesday, May 12.
Abstract Algebra 1: Introduction to group theory
Abstract Algebra 2: When are two groups considered to be the same?
Abstract Algebra 3: An introduction to sets
Abstract Algebra 4: Subsets
Abstract Algebra 5: Introduction to functions
Abstract Algebra 6: How does the output space affect whether a function is onto or not?
Abstract Algebra 7: Composition of functions
Abstract Algebra 8: Why is the composition of two one-to-one functions one-to-one?
Abstract Algebra 9: Function composition is associative
Abstract Algebra 10: The definition of a group
Abstract Algebra 11: The group of nonzero real numbers under multiplication
Abstract Algebra 12: The integers modulo 5 form a group under addition
Abstract Algebra 13: The identity in a group is unique
Abstract Algebra 14: The inverse of any element in a group is unique
Abstract Algebra 15: The inverse of ab is b-1a-1 (socks shoes property)
Abstract Algebra 16: The cancellation law
Abstract Algebra 17: Subgroups
Abstract Algebra 18: Abelian groups
Abstract Algebra 19: Two examples of groups that are not abelian
Abstract Algebra 20: Cyclic groups and subgroups
Abstract Algebra 21: What are the generators of Z/nZ?
Abstract Algebra 22: What are the generators of Z/10Z?
Abstract Algebra 23: Not every group is a cyclic group!
Abstract Algebra 24: Cyclic groups and subgroups are abelian
Abstract Algebra 25: Permutations and permutation groups
Abstract Algebra 26: Identity and inverses in permutation groups
Abstract Algebra 27: The size of the symmetric group Sn is n!
Abstract Algebra 28: What are of the elements of the symmetric group S?
Abstract Algebra 29: How do you write a permutation in disjoint cycle notation?
Abstract Algebra 30: How do you write a product of permutations in disjoint cycle notation?
Abstract Algebra 31: How do you write a product of permutations in disjoint cycle notation?
Abstract Algebra 32: Any permutation can be written as a product of 2-cycles
Abstract Algebra 33: An introduction to the alternating group
Abstract Algebra 34: We should we care about permutation groups?
Abstract Algebra 35: Lagrange's theorem: a subgroup's size divides evenly into the group's size
Abstract Algebra 36: Cosets
Abstract Algebra 37: Cosets drawn via a picture
Abstract Algebra 38: Properties of cosets
Abstract Algebra 39: A proof of Lagrange's theorem
Abstract Algebra 40: Left cosets need not equal right cosets
Abstract Algebra 41: The converse of Lagrange's theorem is not true
Abstract Algebra 42: Introduction to public key cryptography
Abstract Algebra 43: The RSA public key cryptography algorithm
Abstract Algebra 44: A factoring fact related to the RSA cryptography algorithm
Abstract Algebra 45: Fermat's little theorem
Abstract Algebra 46: Why does the RSA algorithm correctly decrypt your message?
Abstract Algebra 47: Introduction to Isomorphisms
Abstract Algebra 48: Which isomorphism class does this group of size four belong to?
Abstract Algebra 49: Definition of isomorphisms
Abstract Algebra 50: An example isomorphism
Abstract Algebra 51: A bijective map that is not an isomorphism
Abstract Algebra 52: Two infinite groups that are not isomorphic
Abstract Algebra 53: Group properties preserved by isomorphisms
Abstract Algebra 54: Ways to tell that two groups are not isomorphic
Abstract Algebra 55: Homomorphisms
Abstract Algebra 56: Example group homomorphisms
Abstract Algebra 57: Examples and non-examples of homomorphisms
Abstract Algebra 58: Kernels of homomorphisms
Abstract Algebra 59: Analogy between group homomorphisms and linear transformations (linear algebra)
Abstract Algebra 60: Quotient groups
Abstract Algebra 61: Quotient group examples
Abstract Algebra 62: Direct products
Abstract Algebra 63: Orders of elements in direct product groups
Abstract Algebra 64: Fundamental theorem of finite abelian groups, Part I
Abstract Algebra 65: Fundamental theorem of finite abelian groups, Part II
Abstract Algebra 66: Introduction to group actions and the orbit-stabilizer theorem
Abstract Algebra 67: The number of rotational symmetries of a soccer ball
Abstract Algebra 68: Stabilizers
Abstract Algebra 69: Orbits
Abstract Algebra 70: The orbit-stabilizer theorem
Abstract Algebra 71: An introduction to fields
Abstract Algebra 72: The finite field of size p, where p is a prime
Abstract Algebra 73: Two equivalent definitions of fields - one condensed, one expanded
Abstract Algebra 74: In a field F, why is 0 times x equal to 0?
Abstract Algebra 75: In a field F, why does xy=0 imply either x=0 or y=0?
Abstract Algebra 76: Rings
Abstract Algebra 77: Subrings
Abstract Algebra 78: The ring of Gaussian integers
Abstract Algebra 79: The ring of all functions from the reals to the reals
Abstract Algebra 80: Zero divisors
Abstract Algebra 81: The cancellation property in integral domains
Abstract Algebra 82: Straight-edge and compass constructions
Abstract Algebra 83: Impossible straight-edge and compass constructions
Abstract Algebra 84: Field extensions for straight-edge and compass constructions
|Date||Book Chapter: Class Topic||Remark|
|Jan 20||Course overview||[Logistics]|
|Jan 22||Course overview|
|Jan 25||Let's meet each other!|
|Jan 27||Chp 1: Preliminaries|
|Jan 29||Chp 1: Preliminaries|
|Feb 1||Chp 3: Groups||Homework due|
|Feb 3||Class got turned into office hours due to technical difficulties||Last day to drop or change grading option|
|Feb 5||Chp 3: Groups|
|Feb 8||Chp 3: Groups|
|Feb 10||Chp 3: Groups and Subgroups|
|Feb 12||Chp 4: Cyclic Groups|
|Feb 15||Snow day - class cancelled (Short Video)|
|Feb 17||Chp 4: Cyclic Groups|
|Feb 19||Chp 4: Cyclic Groups||Homework due|
|Feb 22||Chp 4: Cyclic Groups|
|Feb 24||Chp 5: Permutation Groups|
|Feb 26||Chp 5: Permutation Groups|
|Mar 1||Review break|
|Mar 3||Review break|
|Mar 5||Review break|
|Mar 8||Chp 6: Cosets and Lagrange's Theorem (Video 63-64), (Video 65-67), (Video 68-69)|
|Mar 10||Chp 6: Cosets and Lagrange's Theorem (Video 58), (Video 59-61), (Video 60-62)|
|Mar 12||Chp 7: Cryptography||Homework due|
|Mar 15||Snow day - class cancelled|
|Mar 17||Chp 7: Cryptography|
|Mar 19||Chp 9: Isomorphisms (Video 46-48), (Video 49-51)|
|Mar 22||Chp 9: Isomorphisms (Video 55-56), (Video 52-54)|
|Mar 24||Chp 9: Isomorphisms (Video 56-57), (Direct Products Video 86-89)|
|Mar 26||Chp 11: Homomorphisms (Video 77-80)||Homework due|
|Mar 29||Chp 11: Homomorphisms (Video 80-82), (Video 83-85)|
|Mar 31||Chp 11: Homomorphisms|
|Apr 2||Chp 13: Finite Abelian Groups|
|Apr 5||Chp 14: Group Actions|
|Apr 7||Chp 14: Group Actions|
|Apr 9||Chp 16: Group Actions||Homework due|
Spring Break, Apr 12-16
|Apr 19||Chp 21: Fields|
|Apr 21||Chp 21: Fields|
|Apr 23||Chp 21: Fields|
|Apr 26||Chp 16: Rings|
|Apr 28||Chp 16: Rings|
|Apr 30||Chp 16: Rings||Homework due|
|May 3||Chp 17: Polynomials|
|May 5||Chp 17: Polynomials|
|May 7||Chp 18: Integral Domains|
Wednesday May 12, Homework due