Math 366: Introduction to Abstract Algebra
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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.
Homework
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.
Videos
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
Schedule
| 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 |
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