Model Theory of Several Algebraic Theories
In this chapter we investigate a series of interesting algebraic theories for the properties of completeness, model completeness and quantifier elimination. Not only do we treat the standard examples that have already been frequently treated in the extant literature, but we shall place special value on the theory of valued fields. Since valuation theory does not belong in the standard repertoire of an algebra course, we shall first discuss the necessary concepts and theorems in detail, in Section 4.3. Thereafter we develop special cases (Sections 4.4 and 4.5), and finally the model theory of Henselian valued fields. The goal of this presentation is, among other things, a treatment of a purely number theoretic problem – Artin’s conjecture – in Theorem 4.6.5.
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