## A Course in p-adic AnalysisKurt Hensel (1861-1941) discovered the p-adic numbers around the turn of the century. These exotic numbers (or so they appeared at first) are now well-established in the mathematical world and used more and more by physicists as well. This book offers a self-contained presentation of basic p-adic analysis. The author is especially interested in the analytical topics in this field. Some of the features which are not treated in other introductory p-adic analysis texts are topological models of p-adic spaces inside Euclidean space, a construction of spherically complete fields, a p-adic mean value theorem and some consequences, a special case of Hazewinkel's functional equation lemma, a remainder formula for the Mahler expansion, and most importantly a treatment of analytic elements. |

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### Contents

Projective Limits | 26 |

The Field Qp of padic Numbers | 36 |

Hensels Philosophy | 45 |

The padic Solenoid | 54 |

Exercises for Chapter I | 63 |

Finite Extensions of the Field of padic Numbers | 69 |

Absolute Values on the Field Q | 85 |

Structure of padic Fields | 97 |

Continuous Functions on Zp | 160 |

Umbral Calculus | 195 |

Exercises for Chapter IV | 212 |

Restricted Formal Power Series | 233 |

The Mean Value Theorem | 241 |

The Exponentiel and Logarithm | 251 |

The Volkenborn Integral | 263 |

Exercises for Chapter V | 276 |

Classification of Locally Compact Fields | 115 |

Exercises for Chapter II | 123 |

Definition of a Universal padic Field | 134 |

Special Functions Congruences | 366 |

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### Common terms and phrases

absolute value algebraically closed analytic element assume Banach space binomial bounded canonical choose closed balls coefficients compute consider contains continuous function convergent power series Corollary critical radius defined definition delta operator denote differentiable discrete example exponential extension of Qp field Qp finite extension formal power series formula function f growth modulus hence homomorphism induction inequality infraconnected inverse isomorphism lemma Let f linear locally compact log(l metric space monomials multiplicative neighborhood nonzero notation open ball p-adic integers poles polynomial preceding prime projective limit Proof Proposition proves quotient radius of convergence rational function residue field restricted ring roots of unity satisfies shows sphere subgroup subset subspace surjective Theorem topological group ultrametric field ultrametric space unique unit ball vector space zero