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Full Description
Famous Norwegian mathematician Niels Henrik Abel advised that one should ""learn from the masters, not from the pupils"". When the subject is algebraic numbers and algebraic functions, there is no greater master than Emil Artin. In this classic text, originated from the notes of the course given at Princeton University in 1950-1951 and first published in 1967, one has a beautiful introduction to the subject accompanied by Artin's unique insights and perspectives. The exposition starts with the general theory of valuation fields in Part I, proceeds to the local class field theory in Part II, and then to the theory of function fields in one variable (including the Riemann-Roch theorem and its applications) in Part III. Prerequisites for reading the book are a standard first-year graduate course in algebra (including some Galois theory) and elementary notions of point set topology. With many examples, this book can be used by graduate students and all mathematicians learning number theory and related areas of algebraic geometry of curves.
Contents
General valuation theory
Valuations of a field
Complete fields
$e, f$ and $n$
Ramification theory
The different
Local class field theory
Preparations for local class field theory
The first and second inequalities
The norm residue symbol
The existence theorem
Applications and illustrations
Product formula and function fields in one variable
Preparations for the global theory
Characterization of fields by the product formula
Differentials in $PF$-fields
The Riemann-Roch theorem
Constant field extensions
Applications of the Riemann-Roch theorem
Differentials in function fields
Theorems on $p$-groups and Sylow groups
