Exams
In the Fall of 2011, I gave two finals. Taking the "B" final put a "B" as the absolute cap on the highest grade one could receive in the course. If you ever take a course from me in the future, do not expect a "B" final, as I hope never to repeat this experiment!
- Winter 2011 Final
- Fall 2011 "A" Final
- Fall 2011 "B" Final
- Fall 2015 Final
- Winter 2011 Midterm
- Fall 2011 Midterm
- Fall 2015 Midterm
Assignments
Fall and Winter 2011 all start with groups, then move to rings, and end with fields. In Fall 2015, I did rings first, then fields, then groups. I still believe this is pedagogically correct, but my point of view wasn't well-appreciated by my class!
- Winter 2011 Assignment 1
- Fall 2011 Assignment 1
- Fall 2015 Assignment 1
- Winter 2011 Assignment 2
- Fall 2011 Assignment 2
- Fall 2015 Assignment 2
- Winter 2011 Assignment 3
- Fall 2011 Assignment 3
- Fall 2015 Assignment 3
- Winter 2011 Assignment 4
- Fall 2011 Assignment 4
- Fall 2015 Assignment 4
- Winter 2011 Assignment 5
- Fall 2011 Assignment 5
- Fall 2015 Assignment 5
- Winter 2011 Assignment 6
- Fall 2011 Assignment 6
- Fall 2015 Assignment 6
- Winter 2011 Assignment 7
- Fall 2011 Assignment 7
- Winter 2011 Assignment 8
Notes
These are the notes from Winter '21 from my iPad using Noteshelf. The order is based on Fred Goodman's open text, Algebra: Abstract and Concrete. (Thanks, Fred!).
- Background on Induction and Contradiction
- Background on Proof by Cases
- Symmetries
- Multiplication Tables for Symmetries
- Symmetries and Matrices
- Symmetries of the Square as Matrices
- Introduction to Permutations
- Cycles
- Order of a Permutation and Cycle Decomposition
- Divisibility Of Integers, Part 1 defines divisibility, proves the existence part of the Fundamental Theorem of Arithmetic
- Divisibility Of Integers, Part 2 defines and characterizes the GCD of two integers
- Divisibility Of Integers, Part 3 proves the Euclidean Algorithm
- Divisibility Of Integers, Part 4 proves the uniqueness part of the Fundamental Theorem of Arithmetic
- Equivalence Relations and Modular Arithmetic
- More on Modular Arithmetic
- Counting, Part 1 introduces concepts from combinatorics.
- Counting, Part 2 introduces Euler's Totient Function and properties
- Counting, Part 3 characterizes how the Totient Function behaves with respect to prime factorization
- Introduction to Groups
- Groups and Isomorphisms
- Introduction to Rings
- More on Rings And Fields
- Examples of Fields
- Introduction to Polynomials Over a Field
- Ring-Theoretic Properties of Polynomials
- Divisibility and Irreducibility for Polynomials
- More on Irreducible Polynomials and Divisibility
- Basic Group Theory
- Introduction to Subgroups
- Groups of Small Order
- The Heisenberg Group
- Intersections of Subgroups and CyclicGroups
- Characterizing Cyclic Groups
- Subgroups of Cyclic Groups
- Finite Dihedral Groups
- More on Dihedral Groups
- Introduction to Group Homomorphisms and Examples
- Kernels of Groups Homomorphisms are Normal; the Sign Homomorphism
- The Sign Homomorphism is Well-Defined
- Defining Determinants of Matrices with the Sign Homomorphism
- Cosets and Lagrange's Theorem
- Index For Subgroups
- Quotient Groups
- Homomorphism Theorems for Groups
- Cayley's Theorem and Sofic Groups
- Subrings and Ideals of a Ring
- Quotient Rings and Maximal Ideals
- Types of Rings
- Field Extensions
- Cardinality of Finite Fields
- The Fundamental Field Theorem and Characteristic of a Field
Less Current Notes
These notes were taken by a student in the Winter '11 class and are extremely legible.
- The First Six Lectures -Motivation, Definitions and Examples of Groups, Isomorphism
- Day 7 -Subgroups: Definition and Examples; the Subgroup Tests
- Day 8 -Subgroups Generated by a Subset of a Group
- Day 9 -Cyclic Groups
- Days 10 and 11 -The Center of a Group; Permutations
- Day 12 -Even and Odd Permutations; Cayley's Theorem
- Day 13 -End of Cayley's Theorem; Order of a Group
- Day 14 -Order of an Element and Examples; LaGrange's Theorem
- Day 15 -Cosets: Definition, Examples and the Proof of LaGrange's Theorem
- Day 16 -Normal Subgroups: Definitions and Examples; Homomorphisms
- Day 17 -Kernels of Homomorphisms are Normal, Examples
- Day 18 -Review for Midterm
- Days 19 and 20 -More Midterm Review; Motivation and Definition of Factor Groups
- Day 21 -First Isomorphism Theorem and Applications
- Day 22 -Pictures for Cosets; the Fundamental Theorem of Finite Abelian Groups
- Day 23 -Rings: Definition and Examples
- Day 24 and a repeat of Day 21 -More Examples of Rings; Subring Definition
- Day 25 -Subring Test and Examples of Subrings
- Day 26 -Direct Products of Rings; Ring Homomorphisms and Ideals, Examples
- Day 27 -Maximal and Prime Ideals; Homomorphism Kernel-Ideal Connection
- Day 28 -C_0(R) and its ideals; Zero Divisors
- Day 29 -Types of Rings (Integral Domains, Division Rings, Fields, Euclidean Domains) and Examples
- Day 30 -Principal Ideal Domains and Examples; Euclidean Domains are Principal Ideal Domains
- Day 31 -Fields and Field Extensions
- Day 32 -Characteristic of a Field
- Day 33 -Vector Spaces: Definition and Examples; Degree of a Field Extension
- Days 34 and 35 -Algebraic and Transcendental Elements; Irreducible Polynomials; Fundamental Theorem of Field Theory; Splitting Fields
- Day 36 -Correcting a Glaring Error by the Instructor; Prime Ideals and Eisenstein's Criterion
