Winter 2006

Last updated May 31 2008 12:24:47

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Much as Math 71 is concerned with the study of groups and rings, Math 81 involves the study of fields and more precisely extensions of fields. The theory which is developed along the way provides solutions to a number of classical problems, as well as introduces tools which intertwine the study of fields with that of groups, rings, and vector spaces.

In a sense much of algebra owes its heritage to questions of finding solutions to equations. We all learn in grade school that equations like x^6+ x^5 + ... + x + 1 = 0, have all of their roots in the complex numbers, but what is the "smallest" field containing all its roots, and does that field have arithmetic interest? Is there a formula like the quadratic formula which can describe the roots of all polynomials in terms of radicals and algebraic operations?

Short of resorting to the use of the real or complex numbers which are more germane to analysis, we shall learn how to construct the smallest field containing the rational numbers in which any given polynomial (or all possible polynomials) have their roots. We will show the impossibility of several classical compass and straightedge constructions: trisection of an angle, creating a square with the same area as a circle, or even constructing a cube whose volume is twice that of a given cube. We shall also show that there is no formula for finding the roots of equations of degree at least five using radicals and algebraic operations.

All of these are very pleasing results, owing largely to the distinction between being unable to show something is possible and being able to prove that it is not.