Contents

INTRODUCTION 1

Chapter 1. Class-groups of group-rings 4

1. Horn Descriptions 4

2. The class-group of Z[Q8] 9

3. The second Chinburg conjecture 13

Chapter 2. The evaluation of [X] 17

1. A Horn-description representative of [X] 17

2. Kim's Examples 22

Chapter 3. Quaternion fields over Q2 28

1. Galois actions 28

2. Fundamental 2-extensions 32

3. The fundamental 2-extension in Case A 39

4. The fundamental 2-extension in Case B 45

5. The fundamental 2-extension in Case C 55

Chapter 4. The Invariant in Cases A, B and C 67

1. The strategy of the calculation 67

2. Evaluating [Ker{k')\ in Case A 70

3. Evaluating [Ker[k')} in Case B 80

4. Evaluating [Ker(k')] in Case C 86

Chapter 5. The evaluation of [M] 92

1. The class [M] in Case A 92

2. The class [M] in Case B 112

3. The class [M] in Case C 117

Chapter 6. The conjecture in Cases A, B and C 124

1. Evaluation of [X] in Cases A, B and C 124

2. The proof in Cases A, B and C 127

Chapter 7. Epilogue 128

1. Related and Subsequent Results 128

Bibliography 130

Index 133