Ebook: The Theory of Classical Valuations

In his studies of cyclotomic fields, in view of establishing his monumental theorem about Fermat's last theorem, Kummer introduced "local" methods. They are concerned with divisibility of "ideal numbers" of cyclotomic fields by lambda = 1 - psi where psi is a primitive p-th root of 1 (p any odd prime). Henssel developed Kummer's ideas, constructed the field of p-adic numbers and proved the fundamental theorem known today. Kurschak formally introduced the concept of a valuation of a field, as being real valued functions on the set of non-zero elements of the field satisfying certain properties, like the p-adic valuations. Ostrowski, Hasse, Schmidt and others developed this theory and collectively, these topics form the primary focus of this book.

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In his studies of cyclotomic fields, in view of establishing his monumental theorem about Fermat's last theorem, Kummer introduced "local" methods. They are concerned with divisibility of "ideal numbers" of cyclotomic fields by lambda = 1 - psi where psi is a primitive p-th root of 1 (p any odd prime). Henssel developed Kummer's ideas, constructed the field of p-adic numbers and proved the fundamental theorem known today. Kurschak formally introduced the concept of a valuation of a field, as being real valued functions on the set of non-zero elements of the field satisfying certain properties, like the p-adic valuations. Ostrowski, Hasse, Schmidt and others developed this theory and collectively, these topics form the primary focus of this book.

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In his studies of cyclotomic fields, in view of establishing his monumental theorem about Fermat's last theorem, Kummer introduced "local" methods. They are concerned with divisibility of "ideal numbers" of cyclotomic fields by lambda = 1 - psi where psi is a primitive p-th root of 1 (p any odd prime). Henssel developed Kummer's ideas, constructed the field of p-adic numbers and proved the fundamental theorem known today. Kurschak formally introduced the concept of a valuation of a field, as being real valued functions on the set of non-zero elements of the field satisfying certain properties, like the p-adic valuations. Ostrowski, Hasse, Schmidt and others developed this theory and collectively, these topics form the primary focus of this book.
Content:
Front Matter....Pages i-xi
Introduction....Pages 1-1
Absolute Values of Fields....Pages 3-54
Valuations of a Field....Pages 55-78
Polynomials and Henselian Valued Fields....Pages 79-105
Extensions of Valuations....Pages 107-125
Uniqueness of Extensions of Valuations and Poly-Complete Fields....Pages 127-150
Extensions of Valuations: Numerical Relations....Pages 151-184
Power Series and the Structure of Complete Valued Fields....Pages 185-212
Decomposition and Inertia Theory....Pages 213-226
Ramification Theory....Pages 227-261
Valuation Characterizations of Dedekind Domains....Pages 263-306
Galois Groups of Algebraic Extensions of Infinite Degree....Pages 307-331
Ideals, Valuations, and Divisors in Algebraic Extensions of Infinite Degree over ?....Pages 333-364
A Glimpse of Krull Valuations....Pages 365-372
Back Matter....Pages 373-403

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In his studies of cyclotomic fields, in view of establishing his monumental theorem about Fermat's last theorem, Kummer introduced "local" methods. They are concerned with divisibility of "ideal numbers" of cyclotomic fields by lambda = 1 - psi where psi is a primitive p-th root of 1 (p any odd prime). Henssel developed Kummer's ideas, constructed the field of p-adic numbers and proved the fundamental theorem known today. Kurschak formally introduced the concept of a valuation of a field, as being real valued functions on the set of non-zero elements of the field satisfying certain properties, like the p-adic valuations. Ostrowski, Hasse, Schmidt and others developed this theory and collectively, these topics form the primary focus of this book.
Content:
Front Matter....Pages i-xi
Introduction....Pages 1-1
Absolute Values of Fields....Pages 3-54
Valuations of a Field....Pages 55-78
Polynomials and Henselian Valued Fields....Pages 79-105
Extensions of Valuations....Pages 107-125
Uniqueness of Extensions of Valuations and Poly-Complete Fields....Pages 127-150
Extensions of Valuations: Numerical Relations....Pages 151-184
Power Series and the Structure of Complete Valued Fields....Pages 185-212
Decomposition and Inertia Theory....Pages 213-226
Ramification Theory....Pages 227-261
Valuation Characterizations of Dedekind Domains....Pages 263-306
Galois Groups of Algebraic Extensions of Infinite Degree....Pages 307-331
Ideals, Valuations, and Divisors in Algebraic Extensions of Infinite Degree over ?....Pages 333-364
A Glimpse of Krull Valuations....Pages 365-372
Back Matter....Pages 373-403
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