Instructor: Matthew Cha
Office hours: C237 Wells Hall, WR 2:30p-4:30p
Lecture: A132 Wells Hall, MWR 12:40pm-2:30pm
Email: chamatth "at" msu "dot" edu
This is a first course in abstract algebra and number theory. The topics we will cover include the structure of the integers, congruences, rings, ring homomorphisms, ideals, and quotient rings. This course will emphasize writing mathematical proofs. The prerequisite for this course is MTH 299 or MTH 317H, and it is expected that the student has obtained certain comfort level with writing proofs.
The textbook is T. W. Hungerford, Abstract Algebra: An Introduction, 3rd ed., (Brooks Cole, 2012), however, any edition is fine. We will cover chapters 1-6.
There will be six weekly homework sets. Homeworks will typically be assigned a week before they are due. Solutions will be posted the day after homework is collected. Late homework will not be accepted. Students are allowed to discuss the homework assignments among themselves, but are expected to turn in their own work — copying someone else's is not acceptable. Homework scores will contribute 30% to the final grade. The lowest homework score will be dropped. It is recommended, although not required, that you type your homework solutions in LaTeX. Here is a template you can use: template.tex, template.pdf
There will be one midterm exam, tentatively set for Thursday, June 7th. The final is scheduled for Thursday, June 28; this is the last scheduled day of the course at the regular lecture time. There will be no makeup tests.
The grade for the course will be calculated based on the following percentages: homework 30%, midterm 30%, final 40%. Grades will be posted on D2L.
| 14 May 18 |
§1.1. Arithmetic on ℤ revisited |
| 16 May 18 |
§1.2. Divisibility; greatest common divisor and Euclidean Algorithm HW1 [solutions] (due Wed 23 May 18) |
| 17 May 18 |
§1.3. Primes and Unique Factorization Fundamental Theorem of Arithmetic, primality testing, prime factorization and *cryptography [1] |
| 21 May 18 |
§2.1. Congruence in ℤ and congruence class §14.1. *The Chinese Remainder Theorem HW2 [solutions] (due Wed 30 May 18) |
| 23 May 18 |
§2.2. Modular arithmetic exponents and polynomials, addition and multiplication tables §2.3. Structure of the ℤ mod n; units and zero-divisors §13. Fermat's Little Theorem, *public-key cryptography [1] |
| 24 May 18 |
§3.1. Definition and examples of rings commutative ring, unital ring, matrices with elements in a ring, product of rings, subrings §3.2. Basic properties of rings |
| 28 May 18 | Happy Memorial Day! University closed. |
| 30 May 18 |
§3.2. Units in rings integral domains, fields §3.3. Homomorphisms and isomorphisms HW3 [solutions] (due Wed 6 Jun 2018) |
| 31 May 18 |
§3.3. Properties of homomorphisms §4.1. Polynomial ring R[x] degree of a polynomial, evaluation map |
| 2 Jun 18 | Practice Exam 1 |
| 4 Jun 18 |
§4.1. Division algorithm in F[x] §4.2. Divisors and gcd in F[x] HW4 [solutions] (due Mon 11 Jun 2018) |
| 6 Jun 18 |
§4.2. Euclidean Algorithm for F[x] §4.4. Irreducibles polynomials and the Factor Theorem |
| 7 Jun 18 | Exam 1 [solutions] (please bring your student ID) |
| 11 Jun 18 |
Review of Exam 1 solutions § 4.3. Unique factorization in F[x] HW5 [solutions] (due Mon 18 Jun 2018) |
| 13 Jun 18 |
§ 4.6. *Fundamental Theorem of Algebra § 5.1. Congruence in F[x] § 5.2. The ring F[x]/(p) |
| 14 Jun 18 |
§ 5.3. The field F[x]/(p) when p is irreducible § 5.3. Existence of roots and the extension F ⊂ F[x]/(p) § 6.1. Ideals in R |
| 18 Jun 18 |
§ 6.1. Ideals and congruence in R § 6.2. The quotient ring R/I HW6 [solutions] (due Mon 25 Jun 2018) |
| 20 Jun 18 |
§ 6.2. First Isomorphism Theorem kernel of f, natural homomorphism |
| 21 Jun 18 |
§ 6.2. Third Isomorphism Theorem § 6.3. *Prime and maximal ideals |
| 25 Jun 18 |
Final Exam Review Exam will cover topics from Chps 1-6 (see above) and will not cover topics with * Practice Final |
| 27 Jun 18 |
§ 16.1 *Binary linear codes and symmetric channels § 16.3 *BCH codes |
| 28 Jun 18 |
Final Exam [solutions] I will be in my office Monday 12:40p - 2:30p if you would like to review your graded exam. |
| * - material that will not be tested. |
| [1] |
R. L. Rivest, A. Shamir, and L. Adleman,
A Method for Obtaining Digital Signatures and Public-Key Cryptosystems,
Comm. Assoc. Comput. Mach. 21, 120-126, 1978. [pdf]
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