MAT 4630 - INTRO TO FIELD THRY AND CLSSCL PROBANTIQUITY

credits: 3.0
From the time of ancient Greece through the nineteenth century, two particular mathematical questions drove the creation of large amounts of mathematics. One of these questions was “which numbers are constructible?” That is, using only a compass and a straight edge, can we square a circle, double a cube or trisect an angle? The other question involved solving polynomial equations. It asked for which types of polynomials can we invent a formula similar to the quadratic formula that allows us to find their zeroes? The mathematics used to answer these questions will be traced historically, culminating in the development of field theory and the notion of field extensions in which we find our answers. Objectives: Students who successfully complete this class should: A) Be able to define and use basic terms from field theory including field, field extension, constructible number and solvable by radicals; B) Be able to characterize the set of constructible numbers using field extensions; C) Be able to use this characterization to demonstrate that the three classical problems of antiquity, squaring the circle, doubling the cube and trisecting the angle, have no solutions; D) Be able to characterize the set of polynomials that are solvable by radicals; E) Be able to use a number of important theorems such as Galois’ Theorem, Abel’s Theorem and Hilbert’s Irreducibility Theorem; F) Have improved their ability to work with, understand and communicate higher level mathematics; G) Be able to define mathematical concepts precisely and use these definitions in writing proofs; H) Be able to construct their own original proofs and write these proofs in a logically correct and clear way. Method of Instruction: Lecture and in-class activities. Method of Evaluation: In Class Assignments: 50 points; Homework: 100 points; Midterm Exam: 100 points; Comprehensive Final: 150points; Total: 400 points.

Prerequisite(s): MAT 3050 - LINEAR ALGEBRA