Fields

Abstract

We are now on the brink of solving several of the problems posed in Chapter 1, so it is a good time to review what we know. In Chapter 1 we looked at constructibility by straightedge and compass, and found that the constructible numbers result from 1 by closing under +, −, ×, ÷ (hence they form a field) and square roots of positive numbers. At the same time, we found that the solution of certain geometric problems reduced to the solution of cubic or higher degree equations not obviously solvable by +, −, ×, ÷ and square roots. In particular, duplication of the cube requires solution of the equation x³ − 2 = 0, and construction of the regular p-gon requires solution of the equation x^P−1 + ... + x +1 = 0.