# Spring2017-Math412

MATH 412 Introduction to Abstract algebra II Spring 2017

Instructor Yaping Yang

Office: LGRT 1242

Office Hours: TBA

Email: yaping@math.umass.edu

Class meeting

Math 412 TuTh 8:30-9:45am, LGRT 173

Prerequisites:

An introductory proof course (Math 300 or CS 250), and Math 411.

Course Text:

The text is the same one used for Math 411:

Abstract algebra by Dan Saracino. We will cover rings, fields, and Galois Theory (Sections 16-26).

Homework:

The homework will be collected on days indicated in the course schedule table below. No late homework will be accepted.

All solutions should be written in complete English sentences.

In mathematics, solution of every problem is always a two-step process. First you have this aha moment when you realize that you know how to solve the problem. But don't rush into putting your thoughts to paper. Now you have to convince another person (for example the grader) that you know how to solve the problem. This may require polishing your argument, rearranging its parts, adding extra details and introducing the right notation to make the argument look good on paper.

Each time you use a special proof technique such as proof by induction or proof by contradiction, say so (for example, you may say "I want to argue by contradiction. Suppose x is not equal to zero...").

If you want to use a known theorem or lemma in your homework, simply quote its number from the textbook, for example you may say "By Theorem 16.7, ..." On the test, either state the theorem you are using or refer to it by its name if it has one (for example, you may say "By Lagrange Theorem...", or "By the structure theorem of finitely generated abelian groups...", or "By a theorem I remember from the textbook that the order of k in Zn is equal to n/gcd(n,k)..."

Exams and Quizzes:

There will be two in-class midterms and four 20-minute quizzes. The dates are in the course schedule table below.

The date for the final exam will be determined by the Registrar office.

Two lowest HW grades and one lowest quiz grade will be dropped.

Midterm 1: 20%

Homework: 15%

Midterm 2: 20%

Quizzes: 15%

Final Exam: 25%

Student Presentation: 5%

Course Schedule

Final Exam:

The content of the presentation could be:

**State a theorem, and explain the proof.

**State a theorem, and give one interesting example to explain the theorem

**History of discovery of a theory.

You could either choose one of the following topics, or you could find your favorite topic (related to Abstract algebra).

1) Section 23: Construction with straight edge and compass.(also see Artin: Chapter 13)

2) History of Fermat's last theorem (P213-215)

3) Solving equations of deg n and Galois theory (P 279--281),

4) Localization of a ring, and Quotient field

5) Introduction to elliptic curve cryptography

6)Primes in the ring of Gauss integers (Artin: Chapter11, sec5)

7) Algebraic integers (Artin: Chapter11, sec6)

8) Some Diophantine equation (Artin: Chapter11, sec12)

9) Function fields. (Artin: Chapter13, sec7)

10) Cubic equations (Artin: Chapter14, sec2)

11) Symmetric functions (Artin: Chapter14, sec3)

12) Kummer extension (Artin: Chapter14, sec7)

13) Topology (Artin: Appendix Sec 3)

14) Gauss's 17-sided polygon.

15) Representation theory of the symmetric group

...

Apr.11

8:30-8:40 Matthew Gagnon: History of Fermat's Last Theorem

8:40-8:50 Cole Powers: History of Fermat's Last Theorem.

8:50-9:00 Maral Margossian: History of Fermat's Last Theorem.

9:00-9:10 YangJunqing Qiao: The history/background of Fermat's Last Theorem

9:10-9:20 Lucas Mastro: The life of Eisenstein and his contributions to mathematics

9:20-9:30 Marissa Miller and Portia Anderson: Representation theory of the symmetric group

Apr.13

8:30-8:40 Kexin Jin: Elliptic curves

8:40-8:50 William Dugan: Elliptic curves

(missed) 8:50-9:00 Malik Abdulla: Elliptic curves

9:00-9:10 Christopher Brissette: Introduction to elliptic curve cryptography

9:10-9:20 Brandon Whitchurch: Rubik's cubes

9:20-9:30 Molly O'Neil: Gauss' 17-sided polygon

Apr.20

8:30-8:40 Brandi Mason: Diophantine Equations

8:40-8:50 Wei Xie: Cubic equations

8:50-9:00 Mark Lewis: Gauss primes

9:20-9:30 Douglas Smith: Construction with straight edge and compass.

Apr.25

8:30-8:40 Jordan Esiason: The Monster Group/Fischer-Griess Monster.

8:45-8:55 Samuel Schlesinger: Presentations of groups

9:00-9:10 Connor Wilmot: Solving equations of deg n and Galois theory

9:15-9:25 Nicholas Miller: History of Fermat's Last Theorem

9:30-9:40 Malik Abdulla: Elliptic curves