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| Translator's Introduction | |
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| Introduction | |
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| Notations, Definitions, and Prerequisites | |
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| Principal ideal rings | |
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| Divisibility in principal ideal rings | |
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| An example: the diophantine equations X[superscript 2] + Y[superscript 2] = Z[superscript 2] and X[superscript 4] + Y[superscript 4] = Z[superscript 4] | |
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| Some lemmas concerning ideals; Euler's [characters not reproducible]-function | |
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| Some preliminaries concerning modules | |
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| Modules over principal ideal rings | |
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| Roots of unity in a field | |
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| Finite fields | |
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| Elements integral over a ring; elements algebraic over a field | |
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| Elements integral over a ring | |
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| Integrally closed rings | |
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| Elements algebraic over a field. Algebraic extensions | |
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| Conjugate elements, conjugate fields | |
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| Integers in quadratic fields | |
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| Norms and traces | |
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| The discriminant | |
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| The terminology of number fields | |
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| Cyclotomic fields | |
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| The field of complex numbers is algebraically closed | |
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| Noetherian rings and Dedekind rings | |
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| Noetherian rings and modules | |
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| An application concerning integral elements | |
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| Some preliminaries concerning ideals | |
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| Dedekind rings | |
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| The norm of an ideal | |
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| Ideal classes and the unit theorem | |
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| Preliminaries concerning discrete subgroups of R[superscript n] | |
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| The canonical imbedding of a number field | |
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| Finiteness of the ideal class group | |
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| The unit theorem | |
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| Units in imaginary quadratic fields | |
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| Units in real quadratic fields | |
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| A generalization of the unit theorem | |
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| The calculation of a volume | |
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| The splitting of prime ideals in an extension field | |
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| Preliminaries concerning rings of fractions | |
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| The splitting of a prime ideal in an extension | |
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| The discriminant and ramification | |
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| The splitting of a prime number in a quadratic field | |
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| The quadratic reciprocity law | |
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| The two-squares theorem | |
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| The four-squares theorem | |
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| Galois extensions of number fields | |
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| Galois theory | |
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| The decomposition and inertia groups | |
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| The number field case. The Frobenius automorphism | |
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| An application to cyclotomic fields | |
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| Another proof of the quadratic reciprocity law | |
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| A Supplement, Without Proofs | |
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| Exercises | |
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| Bibliography | |
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| Index | |