ABSTRACT
Chapter 23. A djoining Roots to Polynomials: Splitting Fields
In the previous sections we studied algebraic elements in terms of the poly nomials that they satisfy. Now we reverse the question and ask, how can we determine the roots of a polynomial? This question has several possible in terpretations, and there exists a highly developed theory of approximating roots of a polynomial in a given field. However, we are interested in an ex act algebraic solution, and thus our first interpretation is, “Given / E F[x], can we find a field K that contains all the roots of / ? ” First we locate one root. To understand the question, let us examine a familiar situation.
