Euclidean Quadratic Forms and Adc-forms: I

Introduction 2 1. Normed Rings 4 1.1. Elementwise Norms 4 1.2. Ideal norms 5 1.3. Finite Quotient Domains 5 1.4. Euclidean norms 6 2. Euclidean quadratic forms and ADC forms 7 2.1. Euclidean quadratic forms 7 2.2. Euclideanity 7 2.3. ADC-forms 8 2.4. Euclidean Forms are ADC Forms 8 2.5. The Generalized Cassels-Pfister Theorem 9 2.6. Maximal Lattices 10 3. Localization and Completion 11 3.1. Localization and Euclideanity 11 3.2. Localization and Completion of ADC-forms 12 4. CDVRs and Hasse Domains 13 4.1. Basic defintions 13 4.2. Classification of Euclidean forms over CDVRs 14 4.3. ADC forms over Hasse domains 15 4.4. Conjectures on Euclidean Forms over Hasse Domains 16 4.5. Definite Euclidean forms over Z 16 4.6. Definite ADC forms over Z 17 4.7. Definite ADC forms over F[t] 18 References 19