**AMS Chelsea Publishing**

Volume: 358;
1967;
349 pp;
Hardcover

MSC: Primary 11;

Print ISBN: 978-0-8218-4075-7

Product Code: CHEL/358.H

List Price: $51.00

Individual Member Price: $45.90

**Electronic ISBN: 978-1-4704-3034-4
Product Code: CHEL/358.H.E**

List Price: $51.00

Individual Member Price: $45.90

# Algebraic Numbers and Algebraic Functions

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*Emil Artin*

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.

#### Readership

Graduate students and research mathematicians interested in number theory and algebraic geometry.

#### Reviews & Endorsements

The exposition is (as usual with Artin) quite elegant, and the parallel treatment of number fields and function can be illuminating as well as efficient ... a master of the subject ... It is a true classic in the field.

-- MAA Reviews

Now, after another forty years, and being out of print for the last decades, Artin's classic of timeless beauty has been made available again for new generations of students, teachers, researchers, mathematics historians, and bibliophiles, very much to the benefit of the mathematical community as a whole.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Algebraic Numbers and Algebraic Functions

- Cover Cover11
- Title page iii4
- General preface v6
- Preface vii8
- Contents ix10
- General valuation theory 116
- Valuations of a field 318
- Complete fields 1934
- ๐,๐ and ๐ 5368
- Ramification theory 6479
- The different 86101
- Local class field theory 101116
- Preparations for local class field theory 103118
- The first and second inequalities 127142
- The norm residue symbol 144159
- The existence theorem 165180
- Applications and illustrations 190205
- Product formula and function fields in one variable 213228
- Preparations for the global theory 215230
- Characterization of fields by the product formula 225240
- Differentials in ๐๐น-fields 238253
- The Riemann-Roch theorem 260275
- Constant field extensions 271286
- Applications of the Riemann-Roch theorem 293308
- Differentials in function fields 321336
- Theorems on ๐-groups and Sylow groups 334349
- Index of symbols 339354
- Subject index 343358
- Back Cover Back Cover1366