Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 0 Introductory remarks
- Part I Tools of p-adic Analysis
- 1 Norms on algebraic structures
- 2 Newton polygons
- 3 Ramification theory
- 4 Matrix analysis
- Part II Differential Algebra
- Part III p-adic Differential Equations on Discs and Annuli
- Part IV Difference Algebra and Frobenius Modules
- Part V Frobenius Structures
- Part VI The p-adic local monodromy theorem
- Part VII Global theory
- Appendix A Picard–Fuchs modules
- Appendix B Rigid cohomology
- Appendix C p-adic Hodge theory
- References
- Index of notation
- Subject index
3 - Ramification theory
from Part I - Tools of p-adic Analysis
Published online by Cambridge University Press: 06 August 2022
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 0 Introductory remarks
- Part I Tools of p-adic Analysis
- 1 Norms on algebraic structures
- 2 Newton polygons
- 3 Ramification theory
- 4 Matrix analysis
- Part II Differential Algebra
- Part III p-adic Differential Equations on Discs and Annuli
- Part IV Difference Algebra and Frobenius Modules
- Part V Frobenius Structures
- Part VI The p-adic local monodromy theorem
- Part VII Global theory
- Appendix A Picard–Fuchs modules
- Appendix B Rigid cohomology
- Appendix C p-adic Hodge theory
- References
- Index of notation
- Subject index
Summary
In this chapter, we study the relationship between a complete nonarchimedean field and its finite extensions; this relationship involves the residue fields, value groups, and Galois groups of the fields in question. We distinguish some important types of extensions, the unramified and tamely ramified extensions. We also briefly discuss the special case of discretely valued fields with perfect residue field, in which one can say much more. We introduce the standard ramification filtrations on the Galois groups of extensions of local fields; these will not reappear again until Part IV, at which point they will relate to the study of convergence of solutions of p-adic differential equations made in Part III.
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- Chapter
- Information
- p-adic Differential Equations , pp. 46 - 55Publisher: Cambridge University PressPrint publication year: 2022