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 easilystated 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 x^{2} + y^{2} =
z^{2} 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 nth powers,
with an arbitray natural number n. The nonexistence 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 email 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.
The text has not been chosen yet.
The final grade will be based on class tests (40%),
an individual project (20%)
and a final examination (40%).
Prerequisite:AS/SC/MATH 2022 3.0 or AS/SC/AK/MATH 2222 3.0 or
permission of the Course Coordinator.
