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.

Section 3: 9-10 AM Mondays, Wednesdays, and Fridays in Mason Hall 3330

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

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)

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.

Mathematical Hygiene

Basic proof structures

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

#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

Final: 10:30am-12:30pm Thursday, December 14; place TBA

Quizzes: 25%

Webwork: 5%

Problem Sets: 15%

Midterm Exam: 25%

Final: 30%