In his studies of cyclotomic fields in view of establishing his monumentaltheorem about Fermats last theorem Kummer introduced quotlocalquotmethods. They are co... » 

In his studies of cyclotomic fields in view of establishing his monumentaltheorem about Fermats last theorem Kummer introduced quotlocalquotmethods. They are concerned with divisibility of quotideal numbersquot ofcyclotomic fields by lambda 1 psi where psi is a primitive IpIth root of1 p any odd prime. Henssel developed Kummers ideas constructed the field ofIpIadic numbers and proved the fundamental theorem known today. Kurschakformally introduced the concept of a valuation of a field as being real valuedfunctions on the set of nonzero elements of the field satisfying certainproperties like the IpIadic valuations. Ostrowski Hasse Schmidt and othersdeveloped this theory and collectively these topics form the primary focus ofthis book. TOCAbsolute Values of Fields. Valuations of a Field. Polynomialsand Henselian Valued Fields. Extensions of Valuations. Uniqueness ofExtensions of Valuations and PolyComplete Fields. Extensions of ValuationsNumerical Relations. Power Series and the Structure of Complete ValuedFields. Decomposition and Inertia Theory. Ramification Theory. ValuationCharacterization of Dedekind Domains. Galois Groups of Algebraic Extensions ofInfinite Degree. Ideals Valuations and Divisors in Algebraic Extensions ofInfinite Degree of the Field of Rational Numbers. A Glimpse on KrullValuations. «