ABSTRACT
The three previous chapters showed the impossibility of the three famous Greek constructibility problems. Some of the grubbier parts of the proofs in the previous chapters can be replaced by more elegant arguments-provided we know more about the algebraic structures involved. The next five chapters will develop the machinery needed for these more sophisticated arguments. The proofs we presented in Chapters 36-38 are correct and have the advantage of not needing a great deal of sophistication in order to understand them. However, the arguments to be presented next have the advantage of being much more elegant and concise (at the expense of being less accessible). This is not very surprising as the more we know, the easier it is to express ourselves. As an added bonus, our additional machinery will enable us to prove another impossibility result, regarding the solution of polynomial equations using arithmetic and root extraction.
