Many theorems to be discussed in Chaps. 7 to 10 depend on basic field-theoretic results. This chapter reviews them. Section 6.1 describes the structure of simple fields extensions, obtained from a given field by adjoining a single, new element. Section 6.2 includes the basic results concerning algebraic and finite extensions that will be needed. Section 6.3 introduces the first-order theory \(\textsf {ACF}\) of algebraically closed fields and Sect. 6.4 the extensions of \(\textsf {ACF}\) to fields of positive and zero characteristic. Section 6.5 proves an important theorem about field embeddings, which in turn provides the foundation for our discussion of algebraic closure in the next chapter. The proof makes use of Zorn’s lemma, for which the reader is again referred to Sect. A.6 of the Appendix. Throughout this chapter, we invariably deal with \(\mathcal {L}_{r}\) -structures that model the field axioms. For this reason it is helpful to have a symbolic way of designating multiplicative inverses of non-zero field elements. We write \(1/x\) to refer to the multiplicative inverse of \(x \neq 0\) , while \(y/x\) designates the product of y and the multiplicative inverse of x. These notational choices should not suggest that we are transgressing the linguistic boundaries imposed upon us by the signature of \(\mathcal {L}_{r}\) : they are only special typographical conventions that enable us concisely to identify certain field elements.