Algebraic Numbers and Algebraic Functions (AMS Chelsea Publishing)
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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.
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 Index of symbols Subject index.
- ID: 9780821840757
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