MAT 512

MAT 512 Algebra for Teachers, Spring 2014.

Instructor: Oleg Viro, office 5-110 Math Tower, e-mail: oleg@math.sunysb.edu
Office hours: Monday, Wednesday 4:00pm-5:15pm in 5-110 Math Tower or by appointment.
Grader: Nissim Ranade. E-mail: nissim@math.sunysb.edu
Lectures: Monday and Wednesday, 5:30-6:50pm, in Physics P-127.
Textbook : I.M.Gelfand, A.Shen, Algebra, Birkhauser, 2003
Some parts of the course is covered by additional texts which will be available on the Black Board.
Exams: Final exam Thursday, May 15, 8:30pm-11:00pm,
Course description: The goal of the course is to provide the basic algebraic concepts and tools widely used throughout all mathematical courses both in the high school and college curricula.
Grading system: The final grade is the maximum of the Final exam grade and the weighted average according the following weights: homework 10%, in-class tests 15%, two midterms 20% each, Final 35%.
Homework: will be assigned weekly through the Blackboard and collected in class on Wednesdays. Please write legibly and explain your reasoning clearly and fully. Homework will constitute a significant part of your course grade. Please, be sure to put your name on your homework and staple all pages. You are welcome to collaborate with others and even to consult books, but your solutions should be written up in your own words, and all your collaborators and sources should be listed.
Homework 4 is due on 3/5.
Homework 3 is due on 2/26.
Homework 2 is due on 2/17.
Homework 1 is due on 2/10.
Quizzes: A short quiz (10 min) will be given each week.
Goals: Provide the algebraic material relevant to the everyday work of a high-school teacher both in content and methods of teaching, and form rigorous foundations for arithmetic and algebra of Common Core State Standards.
Students should be able to
- describe the classical number systems used in high school (natural numbers, integers, rational, real and complex numbers) and relations between them;
- use various presentation systems for numbers (e.g., positional systems, aliquot fractions, etc.);
- convert the presentations to each other and perform the arithmetic operations in the systems;
- convert an infinite periodic decimal fraction into an ordinary fraction and back;
- formulate and use the basic properties of algebraic operations (associativity, commutativity, distributivity);
- give definitions of the basic algebraic structures such as monoids, groups, rings, fields, and recognize them in specific situations;
- give definitions of homomorphisms and isomorphisms of groups or rings and recognize whether a specific map matches these definitions;
- give definitions of and fluently operate with the fundamental notions about divisibility of integers (divisor, common divisor, greatest common divisor, prime number, etc.);
- formulate and prove Unique Prime Factorization Theorem, Theorem about division with remainder, Euclidean algorithm;
- find a linear presentation of the greatest common divisor by the Euclidean algorithm, by using continued fractions, by matrix method;
- solve linear Diophantine equations;
- perform basic operations with congruence classes and identify the congruence classes as elements of the residue ring;
- apply modular arithmetics (including the canonical ring homomorphisms) to a wide variety of problems such as divisibility criteria, control of calculations and Diophantine equations;
- identify invertible elements in a residue ring;
- define the Euler function, formulate and prove its properties;
- formulate and prove the Euler theorem and its corollaries (in particular, Fermat's little theorem), as well as apply it to solve congruence problems with large exponents;
- define zero divisor in a ring, define integral domain;
- solve linear equations in the residue rings;
- formulate and prove the theorem about the field of quotients for an integral domain, apply it to the ring of integers;
- prove that polynomials with coefficients in a ring form a ring;
- give definitions of ideal, kernel of a ring homomorphism, quotient ring by an ideal;
- interpret the simplest algebraic extension of fields as quotient rings of the polynomial ring over the field (in particular, represent in this way the field of complex numbers);
- express a symmetric polynomial as a polynomial of elementary symmetric polynomials, use this in problem solving;
- formulate and prove the Vieta theorem.
DSS advisory. If you have a physical, psychiatric, medical, or learning disability that may affect your ability to carry out the assigned course work, please contact the office of Disabled Student Services (DSS), Humanities Building, room 133, telephone 632-6748/TDD. DSS will review your concerns and determine what accommodations may be necessary and appropriate. A written DSS recommendation should be brought to your lecturer who will make a decision on what special arrangements will be made. All information and documentation of disability is confidential. Arrangements should be made early in the semester so that your needs can be accommodated.
Academic integrity statement: Each student must pursue his or her academic goals honestly and be personally accountable for all submitted work. Representing another person's work as your own is always wrong. Faculty are required to report any suspected instance of academic dishonesty to the Academic Judiciary. For more comprehensive information on academic integrity, including categories of academic dishonesty, please refer to the academic judiciary website at http://www.stonybrook.edu/uaa/academicjudiciary/
Critical incident management: Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of Judicial Affairs any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, and/or inhibits students' ability to learn.