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Field Arithmetic explores Diophantine fields through their absolute Galois groups. This largely self-contained treatment starts with techniques from algebraic geometry, number theory, and profinite groups. Graduate students can effectively learn generalizations of finite field ideas. We use Haar measure on the absolute Galois group to replace counting arguments. New Chebotarev density variants interpret diophantine properties. Here we have the only complete treatment of Galois stratifications, used by Denef and Loeser, et al, to study Chow motives of Diophantine statements.
Introduction 1 I. Linear Algebra 4 1. The Linear Space with Real Multiplier Domain 4 2. Finite Dimensional Linear Spaces 10 3. Linear Mappings 13 4. Bilinear and Quadratic Functions 26 5. Multilincar Functions 41 6. Metrization of affine Space 56 II. Differential Calculus 74
