An Introduction to Algebraic Number Theory

1. Front Matter

   Pages i-xi

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2. To the Gauss Reciprocity Law

   • Takashi Ono

   Pages 1-43

3. Basic Concepts of Algebraic Number Fields

   • Takashi Ono

   Pages 44-104

4. Analytic Methods

   • Takashi Ono

   Pages 105-142

5. The lth Cyclotomic Field and Quadratic Fields

   • Takashi Ono

   Pages 143-194

6. Back Matter

   Pages 195-223

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This book is a translation of my book Suron Josetsu (An Introduction to Number Theory), Second Edition, published by Shokabo, Tokyo, in 1988. The translation is faithful to the original globally but, taking advantage of my being the translator of my own book, I felt completely free to reform or deform the original locally everywhere. When I sent T. Tamagawa a copy of the First Edition of the original work two years ago, he immediately pointed out that I had skipped the discussion of the class numbers of real quadratic fields in terms of continued fractions and (in a letter dated 2/15/87) sketched his idea of treating continued fractions without writing explicitly continued fractions, an approach he had first presented in his number theory lectures at Yale some years ago. Although I did not follow his approach exactly, I added to this translation a section (Section 4. 9), which nevertheless fills the gap pointed out by Tamagawa. With this addition, the present book covers at least T. Takagi's Shoto Seisuron Kogi (Lectures on Elementary Number Theory), First Edition (Kyoritsu, 1931), which, in turn, covered at least Dirichlet's Vorlesungen. It is customary to assume basic concepts of algebra (up to, say, Galois theory) in writing a textbook of algebraic number theory. But I feel a little strange if I assume Galois theory and prove Gauss quadratic reciprocity.

• Galois theory
• algebra
• algebraic number theory
• number theory

• The Johns Hopkins University Baltimore, Baltimore, USA

  Takashi Ono

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