330 Information

Mathematics 330 : (Abstract) Algebra
Fall 2024
Introduction

Professor:        Jeff Johannes                           Section 1    MWF 2:30-3:20p    Welles 121
Office:               South 326A
Telephone:     5403 (245-5403)
Office Hours:     Monday 3:30 - 4:30p South 336, Tuesday 4:00 - 5:00p South 336, Wednesday 1:00 - 2:00p Welles 131, Thursday 8:00 - 9:00p South 336, Friday 12:00 - 1:00p South 338, and by appointment or visit
Email Address: Johannes@Geneseo.edu
Web-page:         http://www.geneseo.edu/~johannes

Textbook

A First Course in Abstract Algebra, seventh edition, by John Fraleigh

Overview

Algebra (mathematicians call it simply algebra, and what you did before is precalculus) is about operations. We will explore what operations have in common when working with the integers, the complex numbers, functions, polynomials, matrices, and even transformations of geometric objects. Understanding what we find in common is what algebra is all about.

Learning Outcomes

Students in Math 330, Abstract Algebra, will:

Assess properties implied by the definitions of a group and rings.
Use various canonical types of groups (including cyclic groups and groups of permutations) and canonical types of rings (including polynomial rings and modular rings).
Analyze and demonstrate examples of subgroups, normal subgroups and quotient groups.
Analyze and demonstrate examples of ideals and quotient rings.
Use the concepts of isomorphism and homomorphism for groups and rings.
Produce rigorous proofs of propositions arising in the context of abstract algebra.

Grading

   Your grade in this course will be based upon your performance on homework, edits, examples and three exams. The weight assigned to each is designated below:
        Homework (6)    5% each
        Edits (3)        5% each
        Examples          5%
        In-class exams (2)    15% each
        Final exam (1)          20%

Problem Sets

    There will be six problem sets due on indicated dates. The problems will be mostly proofs. You are encouraged to consult with me outside of class on any questions toward completing the homework. You are also encouraged to work together on homework assignments, but each must write up their own well-written solutions. A violation of this policy will result in a zero for the entire assignment and reporting to the Dean of Students for a violation of academic integrity. A good rule for this is it is encouraged to speak to each other about the problem, but you should not read eachother's solutions. Here is another good idea for this class - if you work with someone, submit different problems from them. Each question will be counted in the following manner:
    0 missing or plagiarised question
    1 question copied
    2 partial question
    3 completed question (with some solution)
    3.5 completed question with only "fixable errors" - minor missteps or minor writing errors
    4 completed question correctly and well-written
Each entire homework set will then be graded on a 90-80-70-60% (decile) scale. Late items will not be accepted.

Solutions and Plagiarism

There are plenty of places that one can find all kinds of solutions to problems in this class. Reading them and not referencing them in your work is plagiarism, and will be reported as an academic integrity violation. Reading them and referencing them is not quite plagiarism, but does undermine the intent of the problems. Therefore, if you reference solutions you will receive 0 points, but you will *not* be reported for an academic integrity. Simply - please do not read any solutions for problems in this class.

Academic Dishonesty

While working with one another is encouraged, all write-ups of assignments must be your own. You are expected to be able to explain any solution you give me if asked. Assignments and exams will be done individually. The Student Academic Dishonesty Policy and Procedures will be followed should incidents of academic dishonesty occur. Any work written, developed, or created, in whole or in part, by generative artificial intelligence (AI) is considered plagiarism and will not be tolerated. While the ever-changing developments with AI will find their place in our workforces and personal lives, in the realm of education and learning, this kind of technology does not help us achieve our educational goals. The use of AI prevents the opportunity to learn from our experiences and from each other, to play with our creative freedoms, to problem-solve, and to contribute our ideas in authentic ways. Geneseo is a place for learning, and this class is specifically a space for learning how to advance our thinking and professional practice. AI cannot do that learning for us.

Opening Meeting

Students will earn two extra points on the first problem set by visiting office hours during the first two weeks of classes, i.e. no later than 9 September.

Edits

    After the first, third, and fifth problem sets are handed in they will be passed to another student (of my sequencing). The editor will not write on the paper of the original author. They will write comments for each problem on the problem set. Focus your editing both on mathematical correctness and mathematical writing, since you will be editing proofs. Edits are due the class after the problem set is due. A student who does not submit a problem set will earn zero for both the problem set and the edit. Edits will be graded on this scale for each problem:
    0 nothing
    1 not useful
    2 did not correct something wrong
    3 did correct something right
    4 no problem
    Problem sets will not be resubmitted, but I hope the edits will be valuable suggestions for the exams and subsequent problem sets. The original author will receive the edits. The editor will receive a score for their edits, and if they want to see the edits again, they will need to track them down from the original author (otherwise we would need two copies, and that seems excessive). There are no edits due before exams in order to allow for a quicker turnaround time.

Examples

    Each student is required to present one example in each third of the course (the exams are dividers). The presentations should be quick, concise, precise, and well-prepared. I will determine priority for presenting examples. Each student who has not yet presented will have priority over students who have presented. A second (or more) example may be presented in order to replace a prior presentation. First attempts take priority over replacements. I have a list of examples that I am looking for. If you are interested in presenting an example not on the list, mention it to me immediately before class. I will be happy to have two examples (if they are different) of each.
    0 no example presented in third, repeat of another's example
    6.5 incoherent attempt
    7.5 dull book example / misses the point - not clear what the intent is
    8.5 interesting book example clearly and insightfully / disappointing own example - not too different
    9.5 own example clearly and insightfully
    10 fascinating own example clearly and insightfully
As with all else, I am happy to consult with you about examples before presenting.

Exams

The exams will consist of a few straightforward problems designed to emphasise a personal understanding of the basics. They will mostly be like the "Exercises" in the text that come before the "Theory" section.

Feedback

Occasionally you will be given anonymous feedback forms. Please use them to share any thoughts or concerns for how the course is running. Remember, the sooner you tell me your concerns, the more I can do about them. I have also created a web-site which accepts anonymous comments. If we have not yet discussed this in class, please encourage me to create a class code. This site may also be accessed via our course page on a link entitled anonymous feedback. Of course, you are always welcome to approach me outside of class to discuss these issues as well.

Social Psychology

Wrong answers are important. We as individuals learn from mistakes, and as a class we learn from mistakes. You may not enjoy being wrong, but it is valuable to the class as a whole - and to you personally. We frequently will build correct answers through a sequence of mistakes. I am more impressed with wrong answers in class than with correct answers on paper. I may not say this often, but it is essential and true. Think at all times - do things for reasons. Your reasons are usually more interesting than your choices. Be prepared to share your thoughts and ideas. Perhaps most importantly "No, that's wrong." does not mean that your comment is not valuable or that you need to censor yourself. Learn from the experience, and always try again. Don't give up.

Accessibility Accommodations

SUNY Geneseo is dedicated to providing an equitable and inclusive educational experience for all students. The Office of Accessibility (OAS) will coordinate reasonable accommodations for persons with disabilities to ensure equal access to academic programs, activities, and services at Geneseo. Students with approved accommodations may submit a semester request to renew their academic accommodations. Please visit the OAS website for information on the process for requesting academic accommodations. Contact the OAS by email, phone, or in-person: Office of Accessibility Services Erwin Hall 22 585-245-5112 access@geneseo.edu

Religious Holidays

It is my policy to give students who miss class because of observance of religious holidays the opportunity to make up missed work. You are responsible for notifying me no later than September 10 of plans to observe the holiday.

Schedule (subject to change)

Date              Topic
August 26     Introduction
        28          Review (0 and 1)
30          2

September 4       3
        6         4

        9        4
        11    5
        13        6 PS 1 due (Sections 2-4)

        16      6 (structure of cyclic groups and rest)
        18       7
        20       8

        23       8, 9 (Cayley and starting orbits)
        25        9    (Even and odd and beyond)
        27    10

        30        Review PS2 due (Sections 5-7)
        2      Review
October 4    XM1

        7          review, 10 (Lagrange)
        9          11
        11       11 Structure of Finite Abelian Groups

    16         13
18         13 Properties of Homomorphism PS3 due (Sections 8-11)

        21         14
        23         14 (pp. 139-141, examples: Fundamental Homomorphism Theorem 14.9/14.11, Normal Subgroup Theorem 14.13, Conjugation and Automorphisms 14.15)
        25       15 (pp. 144-148 Normal subgroups and factor groups 15.2-12)

        28        15 (pp. 148-151 Simple Groups, Maximal normal subgroups. Center, Commutator Subgroup)

        30        18 (pp. 167-172 Ring - check theorem 18.8 properties, Ring homomorphism, Ring isomorphism)
November 1 5.7    18 (pp. 172-174 Field, Subring, Subfield)

        4        19 (pp. 177-181 Zero divisors, Integral domains - and cancelation laws) PS4 due (Sections 13-15)
        6        Review
        8      XM2

        11 discuss / 19 (pp. 181 - 182 Characteristic of a ring)
        13      20 (pp. 184-187 Fermat's Little Theorem, Subgroup of non-zero divisors, Euler's Theorem)
        15       20 (pp. 187-189 Solving congruences)

        18        21 (pp. 190-193 Field of fractions)
        20      21 (pp. 193-196 Field of fractions) PS5 due (Sections 18-20)
        22        22 (pp. 198-202 Ring of Polynomials, Field of rational functions)

        25     22 (pp. 202-207 Evaluation homomorphism, Zeroes of polynomials, rational, and algebraic.)

December 2     23 (pp. 209-214 Polynomial Division Algorithm, Root/factor theorem, Irreducible Polynomials)
      4        23 (pp. 214-218 Theorems for irreducibility, Eisenstein Criterion, Unique factorisation)
        6    Review

        9    Review PS6 due (Sections 21-23)

Thursday, December 12 final exam 12N-3p