Construction Problems and Field Extensions

During the earlier parts of this book, we started always from Euclid’s geometry, developing and expanding it using our modern mathematical awareness. Because of the construction of the field of segment arithmetic, one could even argue that the use of fields in Chapter 4 arises naturally from the geometry. In this chapter, however, we will make use of modern algebra, the theory of equations and field extensions, and in particular the Galois theory, as it developed in the late nineteenth and early twentieth centuries.