This paper details an exploration of fields and field extensions motivated by the solvability of the classic geometric problem of doubling the cube. The reader will compare and contrast the field of numbers generated by a compass and straightedge combination, call it E_s, and the field of numbers generated by paper folding, call it E_o. This project involves building up these fields using the axioms for each that determine what are and are not constructible points and lines. In doing so, we will show that E_o is actually a field extension of E_s; the reader will see why the solution to the doubling of the cube is not constructible with a compass and straightedge and then how simply folding a piece of paper can solve the problem.
Researched and written in the winter of 2016 as a senior thesis.