General information | Course description | Lecture notes | Assignments | Homework schedule | Grading | Links | Fine print

**Lectures:** Tuesday and Thursday, 13:25 – 14:40, Physics Building 119

**Text:** *Abstract Algebra*,
by David S. Dummit and Richard M. Foote (third edition)

**Secondary texts:**

- Artin:
*Algebra* - Lang:
*Algebra* - Herstein:
*Topics in Algebra*

**Name:** Professor Ezra Miller
**Address:** Mathematics Department,
Duke University, Box 90320,
Durham, NC 27708-0320
**Office:** Physics 209
**Phone:** (919) 660-2846
**Email:** *ezramath.duke.edu*
**Webpage:**
*https://math.duke.edu/people/ezra-miller*
**Course webpage:** you're already
looking at it... but it's
*https://services.math.duke.edu/~ezra/501/501.html*
**Sakai site:** available to registered students via Duke NetID
**Office hours:**
Tuesday 14:40 – 16:00 &
Wednessday 14:00 – 15:10, in Physics 209

**Course content:**

- groups
- subgroups
- homomorphisms
- cosets
- quotient groups
- symmetry
- group actions
- permutation representations
- Sylow theorems
- rings
- ideals
- polynomials
- fractions
- modules
- structure theorem over PID

Rings combine two operations: addition and multiplication. In familiar situations, particularly the integers and univariate polynomials, interactions between the two operations lead to fundamental theorems concerning factorization into primes. What results is a main goal of the course: a single structure theorem that classifies all finite abelian groups and also produces Jordan canonical forms of linear transformations.

**Prerequisites:**
Math 501 is a demanding course. Students are expected to have
a firm grasp of linear algebra before beginning the course. In
addition, it is expected that every student begin the course
comfortable and proficient at writing rigorous mathematical
arguments (proofs).

The lecture notes can and often are updated or corrected. If you think you have found an error, check that you have the latest version before sending a correction.

- (notes) Lecture 1: groups, motivation
- (notes) Lecture 2: examples of groups, including fields
- (notes) Lecture 3: uniqueness issues, cyclic groups
- (notes) Lecture 4: homomorphisms, isomorphisms, normal subgroups
- (notes) Lecture 5: equivalence relations, cosets
- (notes)
Lecture 6:
index, Lagrange's theorem, modular arithmetic,
`(ℤ/nℤ)`* - (notes) Lecture 7: groups of small order, products, quotients, first isomorphism thm
- (notes) Lecture 8: group actions, orbits, stabilizers
- (notes) Lecture 9: actions on cosets, counting
- (notes) Lecture 10: finite rotation groups in 3D
- (notes)
Lecture 11:
Cayley's thm, class equation,
`p`-groups, icosahedral group A₅ - (notes) Lecture 12: actions on subsets, Sylow theorems 1, 2, and 3
- (notes) Lecture 13: semidirect products, proofs of the Sylow theorems
- (notes) Lecture 14: groups of order 12
- (notes) Lecture 15: free groups, generators and relations
- (notes) Lecture 16: simplicity of Aₙ, composition series, solvable groups, subgroup lattices
- (notes) Lecture 17: rings: axioms, examples, homomorphisms
- (notes) Lecture 18: ideals, commutative rings, prime & maximal ideals, Chinese remainder theorem
- (notes) Lecture 19: PID and UFD (principal ideal domains and unique factorization domains)
- (notes) Lecture 20: localization
- (notes) Lecture 21: Euclidean domains and Euclidean algorithm
- (notes) Lecture 22: modules and exact sequences
- (notes) Lecture 23: products, coproducts, direct sums, free modules, presentations
- (notes) Lecture 24: rank, submodules of free modules over PID
- (notes) Lecture 25: classification of f.g. modules/PID; applications to abelian groups, Jordan form
- (notes) Lecture 26: coprimary submodules / PID, Nakayama's lemma

**Attendance is required**in lecture, in person. Repeated failure to attend will adversely affect your Homework score.**Participation**, either actively in class or in office hours,**can benefically affect your Homework score**.**Due dates**for the five homework assignments, two midterms, and one cumulative problem set this semester are listed in the table below. You will have roughly two weeks to complete each homework assignment and one week for each exam (midterm or cumulative problem set).**Reading assignments**are listed in the right-hand column of the Assignments table. Ideally, reading for each lecture should be completed before the relevant lecture.- All assignments, including the midterms and cumulative problem
set, will be take-home.
**All are due at noon on the due date unless otherwise announced**. **All solutions you turn in, including midterms, cumulative problem set, and homework, must be typewritten using the provided LaTeX template.**Communicating your ideas is an integral part of mathematics. In addition to the usual PDF files, LaTeX source files for each of the assignments will be provided. You are required to use these LaTeX templates for your solutions by filling in your responses in those files. I am happy to answer any questions you might have about LaTeX, though you should consider asking your classmates first. The reason why**it is crucial for you to use the provided LaTeX templates**is**because they contain margin settings and commands required for grading**. You can add your own macros, but try to ensure they don't disrupt functionality of the grading macros.**Submit your homework, exam, and cumulative solutions using the "Drop Box" feature on Sakai**. Submit your .tex file (the grader or I may comment on your TeX usage) as well as your .pdf file, which can serve as verification that your system produces the same output as ours do and also verification that you submitted the assignment on time, in case there is a problem processing your .tex file (which has happened numerous times in the past).- Late homework will not be accepted except at the discretion of
the instructor. There are, of course, extenuating
circumstances when that discretion is essentially automatic,
such as if you have been ill
**and**you have filed a short-term illness incapacitation form. **If you have technical trouble submitting to Sakai**, then send your files to the instructor by email as a placeholder while the technical problem is diagnosed and resolved.- The logic of a proof must be completely clear for full credit.
- If the course has a grader, then
**the intention is for the the grader to remain anonymous**. If you happen to know the grader's identity, then it is imperative not to contact the grader directly. **Questions about course policies, due dates, grading, and re-grading must be sent to the instructor**. The grader has no responsibility or power over these issues.**You must cite sources in your solutions**. If you rely on so-and-so's theorem, then you must say where you found it. Be specific: "the dual rank theorem" is not precise; in contrast, "[Climenhaga, Theorem 5.10]" is. Theorems are often known by many names, so anyone grading your work could fail to recognize any given theorem by a name you might attach. Note also that**failure to cite properly can sometimes be construed as plagiarism under the Duke Community Standard**.**Students are expected to adhere to the Duke Community Standard**. Students affirm their commitment to uphold the values of the Duke University community by signing a pledge that states:- I will not lie, cheat, or steal in my academic endeavors;
- I will conduct myself honorably in all my endeavors;
- I will act if the Standard is compromised.

**The instructor has a long record of detecting and convicting violations**, resulting in a wide range of sanctions, from grade changes to dismissal from the University. Every case brought by the instructor has resulted in sanctions: I have never lost a case. Don't test me.**Collaboration on homework is encouraged**while you discuss the search for solutions, but when it comes time to write them down, the**written work you turn in must be yours alone**: you are not allowed to consult anyone else's written solution, and you are not allowed to share your written solutions. (It is easy to tell when solutions have been copied or written together.) If you collaborate while searching for solutions,**you must indicate—on the homework page—who your collaborators were**. Failure to identify your collaborators is a breach of the Duke Community Standard.- There is no realistic way the instructor can prevent students
from using an AI tool for homework. However, resorting to an
AI tool before you have thought deeply about a problem robs you
of an opportunity to learn. Repeatedly consulting AI will
substantially lessen your uptake of the course material.
**If repeated reliance on AI results in less than perfect solutions, the instructor reserves the right to deduct more than the nominal value of the relevant items.** - In contrast to the homework policy,
**no collaboration or consultation of human or electronic sources is allowed for either of the two midterms or the cumulative problem set.**You must work completely independently, without giving or receiving help from others or from the internet or any other source except- your lecture notes and previous assignments,
- any lecture notes or other material provided by the instructor, and
- the "Text" and "Secondary texts" listed above.

Assignment | Due Date | Problems | Reading |
---|---|---|---|

Homework #1 | Sat. 9 September | in PDF or LaTeX | §1.1–1.6, §2.1, §2.3 |

Homework #2 | Thu. 21 September | in PDF or LaTeX | §2.2, §2.4, §3.1–3.3, §5.1 |

Midterm 1 | Thu. 28 September | in PDF or LaTeX | §1.7, §4.1, §3.5 |

Homework #3 | Sat. 14 October | in PDF or LaTeX | §4.2–4.6, §5.3–5.5, §6.3 |

Homework #4 | Sat. 4 November | in PDF or LaTeX | §7.1–7.4, §7.6 |

Midterm 2 | Sat. 11 November | in PDF or LaTeX | §8.1–8.3, §7.5 |

Homework #5 | Thu. 30 November | in PDF or LaTeX | §9.1–9.2, §10.1–10.3, §10.5, §12.1, §15.4 |

Cumulative problems | Thu. 7 December, 23:59 | in PDF or LaTeX |

- 50% Homework
- 30% Midterms
- 20% Cumulative problem set

- list of all courses given in the Math Department in Fall 2023
- Official University calendar for academic year 2023–2024
- Duke Community Standard
- Fundamental University definitions and policies concerning academic dishonesty and related matters
- How student conduct issues are resolved

Many thanks are due to Jeremy Martin and Vic Reiner, who provided templates for this webpage many years ago.

The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by Duke University.

*ezramath.duke.edu*

*Thu Dec 7 04:49:17 EST 2023*

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