University of Notre Dame

Mathematics University Seminar

Course Number: MATH 13187 – 1
Instructor: Sonja Mapes Szekelyhidi

One of the most beautiful aspects of mathematics is that sometimes one needs a certain amount of abstraction in order to solve what seems like a fairly simple problem. One of the most wonderful examples of this is the question posed by the ancient Babylonians which can be stated as “given a polynomial p(x) of degree n, can one find a formula for the roots using the coefficients in the polynomial?” Any high school math student knows that the answer for degree 2 polynomials is yes, and that the formula is the quadratic formula. Now the question is what happens for higher degrees?

In this course we will mainly study the notion of symmetry, for example the symmetries of a regular polygon. As it turns out, the study of symmetries of regular polygons is related to figuring out that the answer to the Babylonians’ question is no for polynomials of degree 5 and higher. Alongside investigating the mathematical ideas that go into this answer, we will also venture into the rich history accompanying this story. This will include Renaissance Italian mathematicians challenging each other to public duels where they solve cubic equations, and the ever interesting Frenchman, Evariste Galois, who was mainly responsible for the major breakthroughs in solving this question.

Some of the writing requirement of the course can be fulfilled by writing mathematics homework, but students more interested in history will have the option of writing a more historically based paper concerned with the history of science. Students more interested in mathematics will have the option of fulfilling the writing requirement by investigating the mathematics in the course more deeply than we
will present in class.