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
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