## Math 4281: Introduction to Modern Algebra (Spring 2017)

Lectures: Two sections: MWF 9:05-9:55 in Vincent Hall 113

and MWF 11:15-12:05     in Vincent Hall 6.

You are responsible for attending the section for which you are registered.

Instructor: Gregg Musiker (musiker "at" math.umn.edu)
(To ensure faster responses to emails, please include the course number 4281 in the subject line in email correspondences.)

Office Hours: (In Vincent Hall 251) Monday 10:10-11:00, Wednesday 10:10-11:00, Wednesday 2:30-3:20, or by appointment

## Course Description:

The course will be a basic introduction to fundamental algebraic structures known as groups, rings, and fields. Beginning with familiar structures, such as integers, polynomials, and vector spaces, we will develop the underlying theory that connects these various concepts which will allow us to introduce more exotic examples. We will end this course with a study of symmetries which will illustrate how these different algebraic stuctures are all interconnected.

Along the way, we will establish a few of the fundamental properties satisfied by these algebraic objects and illustrate their importance by looking at applications that follow from their properties. Algebra has a very different feel from calculus and geometry. One starts with a list of allowable axioms that define the algebraic structures and all properties and results flow from these axioms. The material often comes off as abstract; be prepared to put in extra time if this is not your natural inclination. After working with these structues for a while, things become more familiar and natural.

Prerequisites: Math 2283, 3283 or their equivalent. In particular, students will be expected to know some calculus and linear algebra, and have familiarity with proof techniques, such as mathematical induction.

## Textbook:

"Abstract Algebra: A geometric approach" by Theodore Shifrin (1996, Prentice-Hall). The plan is to cover the majority of chapters 1-7 of this textbook.

You might also find useful two textbooks that are available freely (and legally) online:

Thomas Judson's "Abstract algebra: Theory & Applications", Judson book, and

Frederick M. Goodman's "Algebra: Abstract and Concrete"", Goodman book.

Participation in class is encouraged. Please feel free to stop me and ask questions during lecture. Otherwise, I might stop and ask you questions instead.

• Homework (40%): There will be 11 homework assignments, due on Friday of most weeks.

The first homework assignment is due on January 27th.

I encourage collaboration on the homework, as long as each person understands the solutions, writes them up in their own words, and indicates on the homework page their collaborators. Late homework will not be accepted. Early homework is fine, and can be left in my mailbox on the 1st floor. Homework solutions should be well-explained -- the grader is told not to give credit for an unsupported answer. Complaints about the grading should be brought to me.

The score on the lowest homework assignment will be dropped. Solutions will be made available via Moodle

• Three Exams: There will be 3 in-class exams, on Friday March 3rd (15%), Wednesday April 5th (20%), and Friday May 5th (25%).

Each of these will be closed book and closed notes. Since the material will focus on abstract concepts rather than computations, calculators will not be allowed nor necessary. The course material builds on itself which is why later exams are worth more than earlier ones.

Missing an exam is permitted only for the most compelling reasons. You are responsible for obtaining my permission in advance if you must miss an exam. Otherwise you will be given a 0. If you are excused from taking an exam, your other exam scores will be prorated.

• ## Tentative Lecture Schedule (with related reading from Shifrin)

• (Jan 18) Lecture 1: Introduction to the course and cubic equations (Syllabus and 2.4)
• (Jan 20) Lecture 2: Integers, Arithmetic Axioms, and Induction (1.1)
• (Jan 23) Lecture 3: More Induction Examples, the Binomial Theorem, and Primes (1.1-1.2)
• (Jan 25) Lecture 4: The Euclidean Algorithm and the Extended Euclidean Algorithm (1.2)
• (Jan 27) Lecture 5: The Fundamental Theorem of Arithmetic and its proof (1.2)
• (Jan 30) Lecture 6: Modular Arithmetic and Divisibility (1.3, A.3)
• (Feb 1) Lecture 7: Solving Congruences and the Chinese Remainder Theorem (1.3)
• (Feb 3) Lecture 8: Fermat's Little Theorem and the RSA Cryptosystem (1.3)
• (Feb 6) Lecture 9: Selected Solutions to HW 2 and more on the RSA Cryptosystem (Judson 7.2)
• (Feb 8) Lecture 10: The Integers mod n and other examples of Rings (1.4)
• (Feb 10) Lecture 11: Rational Numbers, Integral Domains and Fields (1.4. 2.1)
• (Feb 13) Lecture 12: Polynomial Rings and Euclidean Algorithm revisited (3.1)
• (Feb 15) Lecture 13: Roots of Polynomials I: Division Algorithm and Root Factors (3.1-3.2)
• (Feb 17) Lecture 14: Roots of Polynomails II: Greatest Common Divisors and Unique Factorization (3.2)
• (Feb 20) Lecture 15: Real Numbers and Complex Numbers (2.1-2.3)
• (Feb 22) Lecture 16: More on Complex Numbers and Fundamental Thm of Algebra (3.2)
• (Feb 24) Lecture 17: Proof of the Fundamental Thm of Algebra (3.2-3.3)
• (Feb 27) Lecture 18: Splitting Fields and Integer Polynomials (3.3)
• (Mar 1) Lecture 19: Review for Exam 1 - See Practice Problems on Moodle
As an additional reference: (Goodman 1.6-1.9,1.11)
• (Mar 3) Exam 1
• (Mar 6) Lecture 20: Gauss' Lemma, the Eisenstein Criterion (3.3) and Hints for HW 6
• (Mar 8) Lecture 21: Introduction to Vector Spaces (5.1) and Solutions to Exam 1
• (Mar 10) Lecture 22: Bases, Dimension, and Field Extensions as Vector Spaces (3.2, 5.1)
• (Mar 13) Spring Break
• (Mar 15) Spring Break
• (Mar 17) Spring Break
• (Mar 20) Lecture 23: Finite Fields (5.3)
• (Mar 22) Lecture 24: Finite Fields II (5.3)
• (Mar 24) Lecture 25: Introduction to Groups (6.1)
• (Mar 27) Lecture 26: Introduction to Groups II (6.1)
• (Mar 29) Lecture 27: Subgroups and Group Homomorphisms (6.2)
• (Mar 31) Lecture 28: Isomorphisms and More on Groups (6.2)
• (Apr 3) Lecture 29: Review for Exam 2
• (Apr 5) Exam 2
• (Apr 7) Lecture 30: The Symmetric Group (6.4)
• (Apr 10) Lecture 31: Cosets and Normal Subgroups (6.3)
• (Apr 12) Lecture 32: Cosets and Normal Subgroups II (6.3)
• (Apr 14) Lecture 33: Cosets and Normal Subgroups III (6.3)
• (Apr 17) Lecture 34: Lagrange's Theorem and Groups of Small Size (6.3)
• (Apr 19) Lecture 35: Fundamental Isomorphism Theorem and the Alternating Group (6.3-6.4)
• (Apr 21) Lecture 36: Direct Products Groups (7.5)
• (Apr 24) Lecture 37: Group Actions on a Set (7.1)
• (Apr 26) Lecture 38: The Class Equation and p-groups (7.1, 7.5)
• (Apr 28) Lecture 39: The Alternating Group Revisited
• (May 1) Lecture 40: Glimpse of Galois Theory (2.4, 7.6)
• (May 3) Lecture 41: Review for Exam 3
• (May 5) Exam 3

• ## Homework assignments schedule (tentative)

Assignment Due date Exercises from Shifrin (unless otherwise specified)
Homework 1 Friday January 27 1.1 # 3, 4 (a, b, c, d, g), 6 (a, b)
1.2 # 1 (b,c,d), 4 (a,b), 6, 8
Solutions on Moodle
Homework 2 Friday February 3 1.2 # 7, 15, 16 (b),
1.3 # 3 (For the meaning of "divisibility test", see Prop 3.2 for examples),
5, 20 (a, b), 21 (b)
Solutions on Moodle
Homework 3 Friday February 10 1.3 # 8, 9, 12, 36, 37 (Hint: use problem 36),
39 (a, b)
Solutions on Moodle
Homework 4 Friday February 17 1.4 # 1, 5 (a, b, c, d), 7, 12
Appendix A.3 # 3, 4, 8 (a, b), 9 (a, b, c)
Solutions on Moodle
Homework 5 Friday February 24 1.2 # 11
1.4 # 4, 9
2.3 # 9 (a, c, d, e, g)
3.1 # 1 (a, b, e), 2 (a, d)
Solutions on Moodle
Homework 6 Friday March 10 2.3 # 6 (a, c, d), # 10 (a, b), 17
3.1 # 10 (a, b, e), 13
Solutions on Moodle
Homework 7 Friday March 24 3.2 # 2 (a, b, c), 3 (a, d, i), 4 (a, b), 18
3.3 # 7
5.1 # 11 (b, c, d, e, i), 19
Homework 8 Friday March 31 5.3 # 2, 3 (a,b), 6 (b, c)
6.1 # 1 (a, b, c, d, e, f),
6.2 # 1, 2
Homework 9 Friday April 14 6.2 # 6 (a, b), 13 (a, b)
6.4 # 1 (a, b, c, d), 4 (a), 10 (a, b)
Homework 10 Friday April 21 6.1 # 10 (a, b), 23 (a)
6.3 # 3, 7, 13
6.4 # 12
Homework 11 Friday April 28 6.3 # 20 (a, b)
7.1 # 5, 14 (a, b), 15,
7.5 # 1, 4 (a, b, c), 9