AS/SC/AK/ MATH 3140 6.0

Number Theory and Theory of Equations

      A study of topics in number theory and theory of equations using relevant methods and concepts from modern algebra, such as Abelian groups, unique factorization domains and field extensions.
      Number theory, "the queen of mathematics" (as Gauss called it), is a fascinating subject in which easily-stated problems, understandable to anybody who can add and multiply integer numbers, have occupied amateurs and professionals alike throughout the ages. One of the earliest problems (going back at least 4000 years) must have been that of solving the "Pythagorean" equation x² + y² = z² for integers x, y, z. Presenting the solutions in this case is not very difficult (and we shall deal with it early in the course), but it becomes a famous and very difficult problem if we replace the squares by n-th powers, with an arbitray natural number n. The non-existence of any solutions for n > 2 is known as "Fermat's Last Theorem", a proof of which was found only recently, after centuries of intensive research and with the use of many powerful techniques of modern mathematics.
      Number theory has many modern applications, particularly in cryptography. Any system to secure the flow of potentially sensitive information (encoded on credit cards, or in e-mail communication, for example) makes heavy use of number theory.
      In the course we shall study the predominantly algebraic foundations of number theory and also deal with some modern applications.
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      The final grade will be based on class tests (40%), an individual project (20%) and a final examination (40%).