Local fields and their extensions Book

	Introduction to the second edition
	Foreword to the first edition
Ch. I	Complete Discrete Valuation Fields	1
1	Ultrametric Absolute Values	1
2	Valuations and Valuation Fields	4
3	Discrete Valuation Fields	6
4	Completion	9
5	Filtrations of Discrete Valuation Fields	12
6	The Group of Principal Units as a Z[subscript p]-module	17
7	Set of Multiplicative Representatives	22
8	The Witt ring	26
9	Artin-Hasse Maps	29
Ch. II	Extensions of Discrete Valuation Fields	35
1	The Hensel Lemma and Henselian Fields	35
2	Extensions of Valuation Fields	39
3	Unramified and Ramified Extensions	49
4	Galois Extensions	56
5	Structure Theorems for Complete Fields	61
Ch. III	The Norm Map	67
1	Cyclic Extensions of Prime Degree	67
2	Artin-Schreier Extensions	74
3	The Hasse-Herbrand Function	79
4	The Norm and Ramification Groups	88
5	The Field of Norms	95
Ch. IV	Local Class Field Theory. I	111
1	Useful Results on Local Fields	112
2	The Neukirch Map	123
3	The Hazewinkel Homomorphism	128
4	The Reciprocity Map	139
5	Pairings of the Multiplicative Group	142
6	The Existence Theorem	153
7	Other Approaches to the Local Reciprocity Map	161
8	Nonabelian Extensions	165
Ch. V	Local Class Field Theory. II	171
1	The Multiplicative Group and Abelian Extensions	171
2	Additive Polynomials	179
3	Normic Subgroups	187
4	Local p-Class Field Theory	196
5	Generalizations	203
Ch. VI	The Group of Units of Local Number Fields	207
1	Formal Power Series	207
2	The Artin-Hasse-Shafarevich Map	214
3	Series Associated to Roots	219
4	Primary Elements	228
5	The Shafarevich Basis	232
Ch. VII	Explicit Formulas for the Hilbert Symbol	235
1	Origin of Formulas	235
2	The Pairing (.,.)	241
3	Explicit Class Field Theory for Kummer Extensions	250
4	Explicit Formulas	255
5	Applications and Generalizations	258
Ch. VIII	Explicit Formulas for Hilbert Pairings on Formal Groups	267
1	Formal Groups	267
2	Generalized Hilbert Pairing for Lubin-Tate Groups	272
3	Generalized Hilbert Pairing for Honda Groups	276
Ch. IX	The Milnor K-groups of a Local Field	283
1	The Milnor Ring of a Field	283
2	The Milnor Ring of a Discrete Valuation Field	286
3	The Norm Map	293
4	The Milnor Ring of a Local Field	304
	Bibliography	319
	List of Notations	341
	Index	343