ABSTRACT
A technique which has become very useful in mathematics is that of associating with a given structure a different one, of a type better understood. In this chapter we exploit the technique by associating with any field extension a vector space. This places at our disposal the machinery of linear algebra, leading to a definition of the degree of a field extension. This concept is sufficiently powerful to solve three notorious problems in Euclidean geometry which remained unanswered for over two thousand years. We shall discuss these problems in the next chapter, and devote the present chapter to developing the theory.
