Local FieldsThe p-adic numbers, the earliest of local fields, were introduced by Hensel some 70 years ago as a natural tool in algebra number theory. Today the use of this and other local fields pervades much of mathematics, yet these simple and natural concepts, which often provide remarkably easy solutions to complex problems, are not as familiar as they should be. This book, based on postgraduate lectures at Cambridge, is meant to rectify this situation by providing a fairly elementary and self-contained introduction to local fields. After a general introduction, attention centres on the p-adic numbers and their use in number theory. There follow chapters on algebraic number theory, diophantine equations and on the analysis of a p-adic variable. This book will appeal to undergraduates, and even amateurs, interested in number theory, as well as to graduate students. |
Contents
III | 1 |
IV | 4 |
V | 6 |
VI | 12 |
VII | 16 |
VIII | 18 |
IX | 23 |
X | 26 |
XXXVIII | 189 |
XXXIX | 190 |
XL | 191 |
XLI | 196 |
XLII | 197 |
XLIII | 203 |
XLIV | 208 |
XLV | 211 |
XI | 33 |
XII | 34 |
XIII | 38 |
XIV | 41 |
XV | 53 |
XVI | 59 |
XVII | 64 |
XVIII | 67 |
XIX | 82 |
XX | 84 |
XXI | 87 |
XXII | 88 |
XXIII | 92 |
XXIV | 95 |
XXV | 98 |
XXVI | 105 |
XXVII | 107 |
XXVIII | 114 |
XXIX | 144 |
XXX | 147 |
XXXI | 149 |
XXXII | 165 |
XXXIII | 167 |
XXXIV | 170 |
XXXV | 174 |
XXXVI | 176 |
XXXVII | 178 |
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Common terms and phrases
A₁ abelian extension algebraic closure algebraic extension algebraic number field Appendix archimedean B₁ basis Chapter 9 class number Clearly coefficients complete with respect completely ramified consider converges COROLLARY cubic curve cyclic cyclotomic Deduce defined definition degree denote diophantine equation discriminant DK/k Eisenstein embedding enunciation equivalent Exercise extension K/k function Further galois group given gives global Hasse principle Hence Hensel's Lemma Hint induced irreducible isomorphic Let f(X Let K/k modulo non-arch norm normal notation Note p-adic field p-adic valuation positive integer power series precisely prime divisor prime element proof of Theorem prove quadratic residue class field ring root of unity Show solutions everywhere locally splitting field Strassmann's theorem subgroup Suppose Theorem 1.1 theory topology triangle inequality unit unramified Z-basis zero