Local fields and their extensions Book

	Foreword
	Preface
	Some Conventions
Ch. I	Complete Discrete Valuation Fields	1
1	Ultrametric absolute values	1
2	Valuations and valuation fields	3
3	Discrete valuation fields	5
4	Completion	7
5	Filtrations of discrete valuation fields	10
6	The group of principal units as a Z[subscript p]-module	14
7	Set of multiplicative representatives	17
8	The Witt ring	21
9	Artin-Hasse maps	23
Ch. II	Extensions of Discrete Valuation Fields	29
1	The Hensel Lemma and Henselian fields	29
2	Extensions of valuation fields	32
3	Unramified and ramified extensions	41
4	Galois extensions	47
5	Structure theorems for complete fields	51
Ch. III	The Norm Map	57
1	Cyclic extensions of prime degree	57
2	Artin-Schreier extensions	63
3	The Hasse-Herbrand function	69
4	The norm and ramification groups	76
5	The field of norms	81
Ch. IV	Local Class Field Theory. I	93
1	Complete discrete valuation fields with finite residue field	93
2	The Neukirch map [Upsilon][subscript L/F]	99
3	Functorial properties of [Upsilon][subscript L/F]	104
4	The reciprocity map [Psi][subscript F]	109
5	Pairings of the multiplicative group	113
6	The Existence Theorem	123
7	Hazewinkel's theory and Dwork's theorem	128
Ch. V	Local Class Field Theory. II	135
1	The multipllcative group and abelian extensions	135
2	Additive polynomials	143
3	Normic subgroups	151
Ch. VI	The Group of Units in a p-adic Field	161
1	Formal power series	161
2	The Artin-Hass-Shafarevich map	167
3	Series related to primitive roots	171
4	Primary elements	178
5	The Shafarevich decomposition	183
Ch. VII	Explicit Formulas for the Hilbert Norm Residue Symbol	187
1	Origin of formulas	187
2	The pairing [actual symbol not reproducible]	191
3	Kummer's extensions of a p-adic field	200
4	Explicit formulas	204
5	Applications and generalizations	207
Ch. VIII	Explicit Formulas for the Hilbert Pairing on Formal Groups	213
1	Formal groups	213
2	Generalized Hilbert pairing	217
Ch. IX	The Milnor K-groups of a Local Field	223
1	The Milnor ring of a field	223
2	The Milnor ring of a discrete valuation field	226
3	The norm map	232
4	The Milnor ring of a local field	243
	Appendix A. Absolute Galois group of a local field	255
	Appendix B. Multidimensional local fields	257
	Bibliography	263
	List of Notations	279
	Subject Index	283