ABSTRACT

We are now ready to focus our attention on the problem of whether it is possible to solve all polynomial equations over a subfield of the complex numbers, by ordinary field arithmetic, together with the extraction of roots. We are able to do this in the quadratic case (using the quadratic formula Exercise 9.1), in the cubic case (using the Cardano-Tartaglia approach, Exercise 9.12), and in the quartic case (using the Ferrari approach, Exercise 9.20). In this chapter we will recast this problem in terms of field extensions,

just as we did for the notion of constructible numbers, in Chapter 38. In that context we simplified matters considerably, by focusing our attention on the sequence of ever larger fields necessary to obtain the constructible number. In that case the larger fields were built as quadratic extensions. In our present case, we will need to build larger and larger fields, but allow extensions by higher power roots instead.