TABLE OF CONTENTS Page Numbers Abstract ii Table of Contents iii Acknowledgements iv I. Introduction 1 II. Prerequisites in Logic 3 III. Methods for Limitation of Quantifiers 5 IV. Additive Rational Numbers 9 V. Additive Integers 13 VI. Projection of a Set of Polynomials 19 VII. Real Numbers 27 VIII. Valued Fields 33 IX. Algebraically Closed Fields 35 X. Convergent Power Series Over Complete Algebraically Closed Valued Fields 43 XI. Hensel's Lemma 51 XII. Convergence of Series Solutions of Polynomials 57 XIII. Complete Discretely Valued Fields with Residue Class Field Z p and p Large 73 XIV. Applications to the Ax~Kochen Transfer Principle and the Artin Conjecture 85 Bibliography 89 Notation 9 * Index 9 2
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