Prerequisite: 305.
Distribution: Mathematical Modeling. Majors can fulfill the major presentation requirement in this course in 2005-06.
The main topic of this course is Galois Theory. In 1600 BC the Babylonians were solving quadratic equations. In the sixteenth century, Fontana and Cardano were solving cubic equations. The following natural question arose: which polynomial equations possess radical solutions, i.e. expressions involving roots, addition, subtraction, multiplication and division of complex numbers, and which do not?
In 1832, the French mathematician Evariste Galois proved the landmark result that the general quintic cannot be solved by radicals. His work was not immediately recognized, as he submitted his work several times to the Academy of Sciences in Paris without success. Galois died at the age of 21 in a duel.
In this course
we will study the relation between polynomial solutions, field
extensions and the corresponding Galois groups. We will discuss the
properties of radical extensions and ultimately prove the
unsolvability of the quintic. Throughout the course, we will
investigate the strength of Galois theory in settling various
historic problems: the doubling of the cube, the trisecting of angles
and the construction of n-sided regular polygons. Time permitting,
the course will touch briefly on other classical subjects in Galois
theory, in particular abelian and Kummer extensions.
Prerequisite: 305.
Distribution: Mathematical Modeling. Majors can fulfill the major
presentation requirement in this course in 2005-06.
