In Sect. 3.1 we defined \(\mathcal {L}\) -theory T to be complete iff, for any \(\mathcal {L}\) -sentence \(\phi \) , exactly one of \(\phi , \neg \phi \) is a consequence of T or, equivalently, an element of \(T^{\models }\) . In this chapter we introduce an important test for completeness and apply it to several theories. In particular, we build on the elements of field theory discussed in Chap. 6 to deduce the completeness of \(\mathsf {ACF}_{0}\) and \(\mathsf {ACF}_{p}\) , for each prime number p. The field-theoretic results in this chapter are closely related to the arithmetic of cardinal numbers, covered in the Appendix. We advise the reader unfamiliar with it to study the Appendix, in particular Sect. A.5, before tackling this chapter. The present chapter is structured as follows: in Sect. 7.1 we prove the Löwenheim-Skolem theorems, which jointly yield a completeness test, discussed in Sect. 7.2. Sections 7.3 to 7.7 contain various applications of the test. We look at vector spaces and divisible, torsion-free abelian groups in Sect. 7.3, turn to dense linear orders without endpoints in Sect. 7.4, and finally tackle algebraically closed fields in Sects. 7.5 to 7.7. In Sect. 7.5 we develop a dimension theory for algebraically closed fields. In Sect. 7.6 we prove the existence of algebraic closures and evaluate their size. The knowledge so obtained yields the completeness proofs presented in Sect. 7.7. Section 7.8 closes the chapter with a discussion of model-theoretic algebraic closure and the dimension theory that can be associated with it under special conditions.