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MATH-315 Topics in Algebra
Spring only
The purpose of this course is to introduce students to one of the most beautiful theories of modern mathematics, Galois theory. Developed by Evariste Galois (1811-1832), the aim of the theory was to study solutions to polynomial equations and in particular to determine when a solution to such an equation can be expressed by radicals. The word radicals means a formula that can be built up from the coefficients of the polynomial using the operations of addition, subtraction, multiplication, division and taking roots. Galois’ ground-breaking idea was to associate to any polynomial a certain group and reduce questions to whether the polynomial equation can be solved by radicals or not to
certain properties of the group. The techniques developed were powerful enough to solve a series of problems that puzzled mathematicians since antiquity. Greek geometers were interested in the problem of constructing geometric figures using a ruler and a compass. They knew how to bisect an angle, construct a regular polygon using only these instruments. However, they were unable to show that you can trisect an angle, construct a cube whose volume is twice as that of the unit cube or construct a square whose area was the same as that of a given circle, using only ruler and compass. Galois theory showed that these classical problems have no solution.
After a brief introduction to complex numbers and discussion of the well-known formulas for solutions to cubic and quartic polynomial equations, we will study polynomial rings, fields and field extensions, rule and compass constructions. We will introduce the notions of simple, separable, normal field extensions and finally the Galois extension of fields. The course will end with the fundamental theorem of Galois theory that connects the Galois group of a polynomial to a Galois extension of fields.
This course is for students who have an
interest in problem solving and learning proofs.
Credits: 3
Prerequisites: Math-150, Math-200, Math-215
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More information
Look for this course in the schedule of classes.
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