## Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex MultiplicationModern number theory began with the work of Euler and Gauss to understand and extend the many unsolved questions left behind by Fermat. In the course of their investigations, they uncovered new phenomena in need of explanation, which over time led to the discovery of class field theory and its intimate connection with complex multiplication. While most texts concentrate on only the elementary or advanced aspects of this story, Primes of the Form x2 + ny2 begins with Fermat and explains how his work ultimately gave birth to quadratic reciprocity and the genus theory of quadratic forms. Further, the book shows how the results of Euler and Gauss can be fully understood only in the context of class field theory. Finally, in order to bring class field theory down to earth, the book explores some of the magnificent formulas of complex multiplication. The central theme of the book is the story of which primes p can be expressed in the form x2 + ny2. An incomplete answer is given using quadratic forms. A better though abstract answer comes from class field theory, and finally, a concrete answer is provided by complex multiplication. Along the way, the reader is introduced to some wonderful number theory. Numerous exercises and examples are included. The book is written to be enjoyed by readers with modest mathematical backgrounds. Chapter 1 uses basic number theory and abstract algebra, while chapters 2 and 3 require Galois theory and complex analysis, respectively. |

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### Contents

INTRODUCTION | 1 |

CHAPTER ONE FROM FERMAT TO GAUSS | 7 |

CHAPTER TWO CLASS FIELD THEORY | 97 |

CHAPTER THREE COMPLEX MULTIPLICATION | 199 |

335 | |

343 | |

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### Common terms and phrases

Abelian extension algebraic integer Artin map Artin symbol assume basic class equation class field theory class number coefficients completes the proof complex multiplication complex numbers compute conductor congruence Corollary cubic residues defined discriminant 4n elliptic curves Euler Euler’s conjectures Fermat’s ﬁeld finite form x2 forms of discriminant formula Gal(L/K Gauss genus theory give given Hilbert class field Hint homomorphism ideal class group ideal of OK imaginary quadratic field implies integer solution isomorphism j-invariant K C L ker(X Lagrange lattice Legendre symbol Legendre’s Lemma Let f minimal polynomial modular equation modular function nonzero norm Note number field number theory O-ideal odd prime positive definite positive integer prime ideal prime number prime to f proof of Theorem proper fractional properly equivalent properties Proposition prove q-expansion quadratic forms quadratic reciprocity reduced forms relatively prime represented ring class field splits completely subgroup Theorem 9.2 unique unramified