Jack Jeffries | Math 412 Fall 2017

# Math 412: Abstract Algebra

## Fall 2017 Section 3

#### Syllabus

Math 412 is an introduction to abstract algebra, required for all math majors but possibly of interest also to physicists, computer scientists, and lovers of mathematics. We will begin with ring theory: our first goal is to prove the Fundamental Theorem of Algebra, about the ring you've been studying since elementary school, the integers. In the second half, we will study group theory. In addition to developing many examples, students will prove nearly all statements in this course.
Warning: we differ from the book by including in our definition of ring that every ring contains 1.

#### Sections:

Sections 1 and 2: taught by Professor Karen Smith
Section 3: 9-10 AM Mondays, Wednesdays, and Fridays in Mason Hall 3330

#### Course Assistant:

Yifan Wu

All sections will use the same syllabus, do the same classwork, take the same exams, and do the same homework, regardless of instructor.

#### Prerequisites:

Math 217. Students are expected to know linear algebra and to have had a course in which they have been trained in rigorous proof techniques (induction, proof by contradiction, etc).

#### Course Description:

This class is an introduction to the basic concepts of algebra. The topics covered are approximately Chapters 1-9 and 13 in the textbook. The class is roughly structured as follows: we begin with a rigorous study of arithmetic of the integers (division algorithm, primes, and unique factorization, congruences, modular arithmetic) culminating with the proof of the Fundamental Theorem of Arithmetic. Part II is about basic properties of rings and ring homomorphism (ideals, quotient rings, fields). Here, another important example, which shares many properties of Z, the integers, is the ring of polynomials over a field. Finally, in part III we study the basics of group theory (groups, group homomorphisms, symmetry groups, the symmetric group, normal subgroups, quotient groups, and group actions on sets). The parts are not evenly spaced: I is shorter than II, and II is shorter than III.

#### Office Hours:

My office hours are Mondays 1-3, Wednesdays 12-1, in my office 4827 East Hall
I am available also by appointment if you need me and can't make regular office hours.
Karen's office hours: Mondays 11-12, Wednesdays 1-3, in her office 3074 East Hall
Yifan's office hours: Thursdays 5-6 in the Math Atrium (East Hall second floor)

#### Daily Update:

Wednesday 11/29: Started with a quiz, then did A-C(2) of worksheet #22. Will finish that, then go on to simple groups and structure of finite abelian groups next time.
Monday 11/27: We finished finding the normal subgroups and the quotient groups of the symmetric groups on 4 and 5 elements, front side of worksheet #21. Next time, we will start with the quiz, then go on to the First Isomorphism Theorem for groups, and simple groups. Please read the Diffie-Hellman part of the RSA worksheet / Thanksgiving leftovers worksheet before next week.
Wednesday 11/22: We discussed RSA today, following the RSA worksheet #20 below. We didn't get to Diffie-Hellman. Will resume with normal subgroups after the break.
Wednesday 11/14, Friday 11/16: Discussed normal subgroups and quotient groups, worksheet #19. Monday 11/12: Started with a quiz. Discussed the bonus problem on testing whether an element is a cyclic generator for the multiplicative group of Z/p. Started discussing normal subgroups, following the normal subgroups worksheet. Review sections 8.2 and 8.3, as well as problem D from HW#7 for next time. A summary of what we covered on group actions is posted below.
Friday 11/9: Finished the worksheet on group actions. Discussed the Orbit-Stabilizer theorem and examples. Read 8.3 for Monday.
Wednesday 11/7: Discussed the fact that even though we know multiplicative groups of finite fields are cyclic, what generates them is subtle. Started on the group actions worksheet. Review 8.1 and 8.2 for next time.
Monday 11/5: We reviewed the proof of the theorem that the multiplicative group of any finite field is cyclic. We then defined a group action on a set, and disucssed the example of the integers acting on the real number line. Review the definitions on today's worksheet for next time.
Friday 11/3: Worked on group homomorphisms and Lagrange's theorem. Read 8.2 for next time. Know the main definitions for the quiz.
Wednesday 11/1: Finished classification of small groups. No reading assigned.
Monday 10/30: Started classification of small groups. Read 8.1.
Friday 10/27: We focused on permutation groups today. Did worksheet #15 on permutation groups. If you didn't get through B, look at that, and see if you can do C. Read 7.4 next time. Quiz on Monday.
Wednesday 10/25: Took a quiz, discussed cyclic subroups. Did worksheet #14 on subgroups and generators. Do up through C if you didn't get through it in class. Read 4.5 for next time.
Monday 10/23: Collected some examples of groups: additive groups of rings, multiplicative groups of units, and groups of symmetries/permutations. Did worksheet #13 on Groups 2. Altogether, we found evidence for the conjecture that the order of any element divides the order of the group. Read 7.3 for next time. There is a quiz on Wedensday.
Friday 10/20: Did worksheet #12 on Groups. Read 7.1 and 7.2 for next time. No quiz on Monday.
Wednesday 10/18: Reviewed what elements are units in quotients of polynomial rings, and how to solve linear equations there. Showed that the quotient of a polynomial ring over a field by an irreducible polynomial is a field, and used the quotient ring construction to construct fields where polynomials have roots.
Friday 10/13: Discussed the First Isomorphism Theorem.
Wednesday 10/11: Continued with quotient rings, and introduced the First Isomorphism Theorem.
Monday 10/9: Started to discuss quotient rings. Showed that addition and multiplication are well-defined. Read 6.2 for next time.
Friday 10/6: Division algoithm in general. Ideals in polynomial ring, both over fields and other rings.
Wednesday 10/4: We reviewed some of the peculiarities about the defintions of ideals/subrings. We talked a bit about polynomial rings and the division algorithm for polynomial rings over fields. Got through C on worksheet #11. Will continue with polynomial rings and ideals on Friday. Read 4.3.
Monday 10/2: Started talking about ideals, and generators of ideals on worksheet #10. We showed that every ideal in the ring of integers is generated by one element. More ideals Wednesday. Read 4.1 and 4.2.
Friday 9/29: Discussed ring homomorphisms and did A, B from worksheet #9. Discussed C and F briefly. Defined ideal. Note that an ideal is NOT a subring in terms of our definition of ring, because 1 is usually not in an ideal.
Wednesday 9/27: Did worksheet #8 on Rings 2. On Friday, we discuss homomorphisms. Monday 9/25: Did worksheet #7 on Rings. Will continue talking about rings on Wednesday. Read section 3.2.
Friday 9/22: Went through worksheet #5 on Systems of Congruences, and then did worksheet #6 on operations. Reread Section 3.1 for Monday.
Wednesday 9/20: Did worksheet #4 on solving the equation [a]x=[b] in Z_N. We all discussed through Problem B, and Problem C1, which said solutions may not exist and may not be unique if N is not prime. Read 14.1 and 3.1 for Friday. Monday 9/18: Did Quiz #2, and and mostly finished the Congruence worksheet. Read Section 2.3 and start 14.1 for Wednesday.
Friday 9/15: Did worksheet #3 on Congruence up through part D. Will continue with E and F on Monday. Finish reading section 2.2 for Monday.
Wednesday 9/13: Started with quiz #1, then discussed equivalence relations and equivalence classes. Read 2.1 and start 2.2 for Friday.
Monday 9/11: Did worksheet #2 on the Fundamental Theorem of Algebra. We recapped through part D. You may want to go over part E in the solutions. Read Appendix D and start 2.1 for Wednesday.
Friday 9/8: Did worksheet #1 on the Euclidean algorithm. We discussed how to find the gcd of two numbers as a linear combination, why the Euclidean algorithm gives a proof of this fact, and the difference between a constructive proof and a nonconstructive proof. Dicussed Theorem 1.4 of Section 1.2. Review 1.2 and read 1.3 for next time.
Wednesday 9/6: We discussed syllabus matters, loosely discussed the notion of a ring, and covered Section 1.1, inculding the proof of the division algorithm in detail. Read Sections 1.1 and 1.2 for next time. Know definitions of divides, GCD, and go over proof of Theorem 1.2.

#### Required Text:

Abstract Algebra: an introduction by Thomas W. Hungerford, 3rd edition (earlier editions are OK but homework numbering and page numbers may differ).

#### Some review materials:

The Joy of Sets
Mathematical Hygiene
Basic proof structures

#### Homework:

Problem Set 11 due December 8
Problem Set 10 due December 1
Problem Set 9 due November 17
Problem Set 8 due November 10
Problem Set 7 due Friday November 3. Typo on E3 corrected on Sunday.
Problem Set 6 due Friday October 27.
Problem Set 5 due Friday October 13. Some typos corrected on Sunday; please check you solved the current version of D. Please turn in A-B and C-E on separate pages.
Problem Set 4 due Friday October 6.Solutions
Problem Set 3 due Friday Sept 29.
Problem Set 2 due Friday Sept 22.
Problem Set 1 due Friday Sept 15. Solutions

#### Worksheets:

#23 First Isomorphism Theorem for Groups
#22 Thanksgiving Leftovers
#21 RSA and Diffie-Hellman
#20 Normal subgroups and quotients
#19 Normal subgroups
#18. Group Actions on Sets Answers
#17. Homomorphisms and Lagrange's Theorem Solutions to back
#16. Classification of groups Solutions
#15. Symmetric Groups Solutions
#14. Subgroups and generators Solutions
#13. Groups 2 Solutions
#12. Groups
#11. Polynomial rings and ideals Solutions
#10. Ideals
#9. Homomorphisms
#8. More rings
Correction: The ring with "hearts" multiplication has a different addition. Here are correct + and times tables for the two rings.
#7. Rings Solutions
#6. Operation! Solutions
#5. Systems of Congruences
#4. [a]x=[b] Solutions
#3. Congruence Solutions
#2. Fundamental Theorem of Algebra Solutions
#1. Euclidean Algorithm Solutions

#### Quizzes:

Quizzes with solutions

#### Course Expectations:

Math 412 students are responsible for learning the material on their own through individual reading of the textbook before coming to class. Like in Math 217, you will often work together on more theoretical concepts in small groups using worksheets in class. You will be expected to work out more computational exercises on your own, which will be supplemented with some webwork when possible. You will also have a graded, written problem set (think Math 217 Part B) due Fridays. Attendance is required. There will be two exams (one midterm and a final). There will be many quizzes, some on the reading.

#### Exams:

Final: 10:30am-12:30pm Thursday, December 14; CCL1528
Review sheet
Notes on group actions
Some True/False questions
Midterm: 6:30pm-8:30pm Wednesday, October 18; NS 2140

#### Review Session:

Sunday at 1pm in EH3866