Math 433- Applied Algebra
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2017/4/27, 13:30 Please complete the evaluations You can complete them in an electronic device of your choice (smartphone, laptop, or tablet). You open a browser and follow these three simple steps: Go to http://math.tamu.edu Login with their NetID and password. Begin. Deadline is 7:00am Thursday, May 4th. I will really appreciate your feedback. Thank you for your time! Lecture 04/27 Exam III Exam III is comprenhensive. It covers chapters 1, 2, 3, 4.3, 5, 6, 8 and 9.1, 9.2 (only the part covered in class from section 9.2). Date: 04/27 from 2:20 to 3:35 pm at Blocker 164. Please bring your id, pen or pencil and eraser. Lecture 04/25 Section 9: Group homorphisms: definition and examples. Kernel and image of group homomorphisms. Isomorphisms: definition and example. Normal subgroups: definition and examples. Index $2$ subgroups are normal. Section 9.1: Direct products: definition, examples and characterization (Theorem 9.19). 2017/04/24, 16:56 Homework 11 (will not be graded, it is for you to practice). I might give you some extra exercises to practice after class on Tuesday 04/25. Section 8.2 (page 132): 1, 2, 5. If you want extra practice try 4, 6. Section 9.2 (page 143): 4, 5, 6, 7. 2017/04/20, 16:16 Homework 10 was the last graded homework. The exercises of sections: 8.2 and chapter 9 are very useful as practice for the exam but will not be graded. 2017/04/20, 16:13 Extra Office hours on Monday April 24. 3-4 pm at Blocker 510A (in addition to the regular ones that week on Tuesday and Wednesday). Lecture 04/20 Section 8.2.2: RSA encryptation system: how it works, encryption and decryption keys. Relation with Euler's Theorem and the units of Z_n. Section 9: Group homorphisms: definition and examples. Kernel and image of group homomorphisms. Isomorphisms: definition and example. 2017/04/18, 18:36 Updated version of Homework 10. Due date: 04/20. Homework will be collected in the lecture. From the textbook: Section 8.1 (page 129): 9, 10, 11, 12, 13, 15, 16. If you want extra practice try 8, 14, 17, 18. Lecture 04/18 Section 8.1.2: Cosets of a subgroup. Lagrange's Theorem and some applications. Euler's Theorem (and Fermat's Little Theorem). Examples. Lecture 04/13 Section 8.1: Units of Z_n and Euler phi function. Subsection 8.1.1 Subgroups: definition, characterization, examples. Cyclic subgroups: definition, characterization, examples. 2017/04/14, 17:16 Homework 10 now available. Due date: 04/20. Homework will be collected in the lecture. From the textbook: Section 8.1 (page 129): 9, 10, 11, 12, 13, 15, 16. If you want extra practice try 8, 14, 17, 18. Section 8.2 (page 132): 1, 2, 5. If you want extra practice try 4, 6. Lecture 04/11 Section 6.4: Constructible angle. Squaring a circle. Reading assigment: Angle trisection and Constructible polygons. Section 8.1: Definition of a group. Examples coming from rings: Group of units of a ring; the ring with its additive structure is a group. Basic properties of groups: uniqueness of identity and inverses. Cyclic groups and generators. Example sof cyclic groups: (Z, +); (Z_n, +); untis of Z; groups of cardinal 2 and 3. Lecture 04/06 Section 6.3: Constructible field, the plane of a constructible field. Coordinates of points produced by the basic constructions, case of the intersection of two lines in the plane of a constructible field. Dimension of a field extension by a root of a quadratic polynomial over the rational numbers. Construcibility criterion using the dimension of field extensions. Section 6.4: Constructible angle. Comments on angle trisection. Duplicating a cube. Lecture 04/04 Exam II 2017/03/31, 11:56 Office hours are cancelled on Tuesday 04/04 and Wednesday 04/05. 2017/03/31, 11:55 Exam II covers sections 4.3-6.2 (inclusive). Date: 04/04 from 2:20 to 3:35 pm at Blocker 164. Please bring your id, pen or pencil and eraser. Extra Office hours on Friday March 31. 09-10 am at Blocker 510A. Extra Office hours on Wednesday March 29. 1-2 pm at Blocker 510A (in addtion to the regular ones from 2-3 pm). 2017/03/28, 18:56 Updated version of Homework 8. Due date: 03/30. Homework will be collected in the lecture. From the textbook: Section 5.4 (page 90): 8, 9, 12,, 14, 15, 16, 17, 18. If you want extra practice try 10, 11, 13. Chapter 6 (page 94): 1, 2(a), 2(b). If you want extra practice try 2(c), (d). Extra Office hours on Monday March 27. 1:30-2:30 pm at Blocker 510A. Lecture 03/23 Section 5.4: Connection between F[x]/I and F(alpha). Algebraic and trascendental elements in a field. Examples. Minimal polynomial of alpha over F. Field homomorphisms. Dimension of the extension F(alpha) of F and degree of the minimal polynomial of alpha. Examples. Characterization of algebraic elements in a field. Lecture 03/21 Section 5.3: Field extensions. F[x]/(f) field extension of F if and only if f is irreducible. Examples. Dimension of a field extension. Field extensions given by adjoining an element (single extension, and more). Tower of field. Dimension formula. Lecture 03/16 Spring break - No lecture. Lecture 03/14 Spring break - No lecture. Lecture 03/09 Section 5.2: Cosets of an ideal, conditions for two cosets to coincide, operations between cosets: addition and product, the ring R/I. Irreducible elements in a commutative ring. Examples: Z, F[x]. Units in F[x]. Specific examples for F[x] (for different fields F). Division algorithm and quotient rings F[x]/I. Examples. Section 5.3: Field extensions: definition. 2017/03/07, 17:16 Updated Homework 6. Due date: 03/09. Homework will be collected in the lecture. From the textbook: Section 5.1 (page 76): 2, 3, 5, 6, 9, 10. If you want extra practice try 1, 7, 8, 11. Section 5.2 (page 83): 1, 2. If you want extra practice try 3, 4. Comment: Exercise 5, Section 5.2 will be part of Homework 7 but Exercise 9, Section 5.2 will not be part of next homework. Lecture 03/07 Section 5.2: Ideal generated by certain elements, principal ideal, Principal Ideal Domain (PID). The ring F[x] is a principal ideal domain (where F is a field) but F[x,y] is not. Cosets of an ideal, conditions for two cosets to coincide, operations between cosets: addition and product. 2017/03/03, 17:56 Homework 6 now available. Due date: 03/09. Homework will be collected in the lecture. From the textbook: Section 5.1 (page 76): 2, 3, 5, 6, 9, 10. If you want extra practice try 1, 7, 8, 11. Section 5.2 (page 83): 1, 2, 5, 9. If you want extra practice try 3, 4, 6. Lecture 03/02 Review of Section 4.3: Properties and relations of generator matrices, parity check matrices and dual codes. Section 5.1: Arithmetic of polynomials. Degree of a polynomial, monic polynomials. Divisibility of polynomials and greatest common divisor. Division algorithm of polynomials; gcd as linear combination. Section 5.2: Ideals: definition, examples and properties. Tuesday February 28 is cancelled (sewage back-up problem in the first floor of Blocker - there will be no water and facilities). Office hours on Tuesday Febraury 28 are the usual ones: 3:45-4:45 pm at Blocker 510A. Lecture 02/23 Section 4.3: Linear codes: examples (span of a set), relation between distance of the code and weight of the codewords. Generator matrix: properties and example. Parity check matrix. Dual code. Properties and relations of generator matrices, parity check matrices and dual codes. Lecture 02/21 Exam I Exam I covers sections 1.1-4.2 (inclusive). Date: 02/21 from 2:20 to 3:35 pm at Blocker 164. Please bring your id, pen or pencil and eraser. 2017/02/17, 18:55 Homework 5 now available. Due date: 03/02. Homework will be collected in the lecture. From the textbook: Section 3.2 (page 50): 5, 10, 11, 12, 13, 14, 19, 21, 22, 24. If you want extra practice try 15, 16, 17, 18, 20, 23, 25. Section 3.3 (page 55): 1, 2, 3, 5, 6, 7. If you want extra practice try 4, 8. Section 4.3 (page 71): 1, 2, 3, 4, 5. Lecture 02/16 Section 3.2: First properties of rings: zero divisors: definition and examples, cancellation law for multiplication, units: definition, examples, and properties. Section 3.3: Fields: definition, examples: Z_n, Q(i), and more. Integral domains: definition, finite integral domains are fields. Reminder-Early Feedback I would really appreciate if you can take some time to give me some Early Feedback. It will be great if I can improve the lectures and you can make the most of this course. You just need to go to Pica website to and enter the Student Login at your earliest convenience. The deadline for submission of feedback is 11:59:59 PM Sunday, February 19th. Extra Office hours on Monday February 20. 1-2 pm at Blocker 510A. 2017/02/15, 13:55 Updated version of Homework 4. Due date: 02/16. Homework will be collected in the lecture. From the textbook: Section 3.1 (page 50): 1, 3, 4, 6, 7. If you want extra practice try 2, 5, 8, 9. Reading assignment: Sections 4.1 and 4.2 Section 4.1 (page 61): 1, 2, 3, 4. If you want extra practice try 5. Section 4.2 (page 66): 1, 2, 3, 5, 6. If you want extra practice try 4, 7. Extra Office hours on Wednesday February 15. 1-2 pm at Blocker 510A. Lecture 02/14 Quiz 2. Section 3.1: More examples: ring of polynomials, cartesian product of rings. Section 3.2: First properties of rings: uniqueness of the additive identity and the additive inverse, cancellation law for addition, some properties of the additive inverse and additive identity. 2017/02/09, 17:45 Homework 4 now available. Due date: 02/16. Homework will be collected in the lecture. From the textbook: Section 3.1 (page 50): 1, 3, 4, 5, 10, 11, 12, 13, 14, 19, 21, 22, 24. If you want extra practice try 2, 6, 7, 8, 9, 15, 16, 17, 18, 20, 23, 25. Reading assignment: Sections 4.1 and 4.2 Section 4.1 (page 61): 1, 2, 3, 4. If you want extra practice try 5. Section 4.2 (page 66): 1, 2, 3, 5, 6. If you want extra practice try 4, 7. Lecture 02/09 Section 2.4: Rewiew of main concepts. Section 3.1: Cartesian product, binary operations. Definition of ring and examples. Lecture 02/07 Quiz 1. Section 2.3: Hamming code: Review of the Hamming matrix and its properties. Review of the decoding algorithm. Perfect codes. Section 2.4: Coset decoding: Review of definition of coset, equivalence relation, cosets are equivalence classes, properties. Example. Decoding algorithm: syndrome, error word and coset leader. Example. 2017/02/03, 17:10 Homework 3 now available. Due date: 02/09. Homework will be collected in the lecture. From the textbook: Section 2.1 (page 27): 4, 5, 6. Section 2.3 (page 35): 1, 2, 3, 4, 5. Section 2.4 (page 38): 1, 2, 3. If you want extra practice try 4. Section 2.5 Reading assignment: please read this section and solve the exercises (page 40): 1, 2, 3. If you want extra practice try 4. Lecture 02/02 Section 2.3: Hamming code: Hamming matrix, properties, codewords. Decoding algorithm. Section 2.4: Coset decoding: definition of coset, equivalence relation, cosets are equivalence classes, properties. 2017/01/31, 17:20 Updated version of Homework 2. Due date: 02/02. Homework will be collected in the lecture. From the textbook: Section 1.2 (page 16): 18, 19, 21, 25. If you want extra practice try 15, 16, 17, 20, 22. Section 1.3 (page 20): 2, 3, 5, 6, 7. If you want extra practice try 1, 4, 8. Section 2.1 (page 27): 1, 2, 3. If you want extra practice try 4, 5, 6. Section 2.2: Reading assignment: please read this section and review from your lecture notes from Linear algebra. You need to recall some of the basic results about matrices, such as Gaussian elimination, rank, nullity. Read example 2.17. Lecture 01/31 Section 2.1: Review of weight and distance, properties and examples, MLD (maximum likelihood detection), t-error correcting codes (error correction capability). Examples. Error correction capability of a code C of distance d. Lecture 01/26 Section 1.2.2: Euclidean algorithm (continuation). How to get the integer linear combination using this algorithm. Examples. Section 1.3: Error detection with identification numbers. Recall the 3 main examples from section 1.1. Single digit errors; Transposition errors. Discuss both for the previous examples. Design of scheme that detects every single digit error. Section 2.1: Error correcting codes. Basic notions: words, linear codes, codewords. Examples. Definition of weight wt and distance D. Some properties. Modification in Homework 1 (some of the exercise of Section 1.2 will be part of Homework 2). Due date: 01/26. Homework will be collected in the lecture. From the textbook: Section 1.1 (page 4): 1(a), 2, 3, 4. If you want extra practice try 1(c), 5, 6. Section 1.2 (page 8): 1, 2, 4, 5. If you want extra practice try 3, 6. Section 1.2 (page 16): 2, 3, 4, 5, 6, 7, 12, 13. If you want extra practice try 1, 9, 11. Lecture 01/24 Section 1.2: Modular arithmetic: some differences between Z_n and Z (units and zero divisors). Section 1.2.2: Greatest common divisor (gcd): definition, basic properties, examples. Integer linear combination and gcd: relation and properties. Prime and relative prime numbers. Euclidean algorithm. 2017/01/21, 17:00 Homework 2 (Part 1) now available. Due date: 02/02. Homework will be collected in the lecture. From the textbook: Section 1.3 (page 20): 2, 3, 5, 6, 7. If you want extra practice try 1, 4, 8. Section 2.1 (page 27): 1, 2, 3. If you want extra practice try 4, 5, 6. Section 2.2: Reading assignment: please read this section and review from your lecture notes from Linear algebra. You need to recall some of the basic results about matrices, such as Gaussian elimination, rank, nullity. 2017/01/19, 20:15 Extra Office hours on Friday January 20. 11 am-12 pm at Blocker 510A. 2017/01/19, 20:00 Homework 1 now available. Due date: 01/26. Homework will be collected in the lecture. From the textbook: Section 1.1 (page 4): 1(a), 2, 3, 4. If you want extra practice try 1(c), 5, 6. Section 1.2 (page 8): 1, 2, 4, 5. If you want extra practice try 3, 6. Section 1.2 (page 16): 2, 3, 4, 5, 6, 7, 12, 13, 18, 19, 21, 25. If you want extra practice try 1, 9, 11, 15, 16, 17, 20, 22. Lecture 01/19 Section 1.2: Last part of the proof of Division Algorithm (uniqueness). Definition of congruence module n, properties, equivalence classes and definition of Z_n. Relation between congruence module n and remainders under division by n, cardinality of Z_n. Arithmetic operations in Z_n Section 1.2.1: Definition of addition and multiplication on Z_n, examples and table of addition and multiplication in Z_n, correcteness of the definition of the operations and properties. Lecture 01/17 Explanations of the course contents and syllabus. Section 1.1: Examples of identification numbers: ISBN-10 (Section 1.1.3). Reading assigment: USPS Zip Code (Section 1.1.1), Universal Product Code (Section 1.1.2), ISBN-13 (Section 1.1.3). Section 1.2: Definition of divisibility, Division Algorithm. |