Algebraic Independence

Abstract

In this chapter, we discuss the structure of an arbitrary field extension F < E. We will see that for any extension F < E. there exists an intermediate field F < F(S) < E whose upper step F(S) < E is algebraic and whose lower step F < F(S) is purely transcendental, that is, there is no nontrivial polynomial dependency (over F) among the elements of S, and so these elements act as “independent variables” over F. Thus, F(S) is the field of all rational functions in these variables.