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 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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Abelian extension Artin map assume ax² basic biquadratic C(OK class equation class field theory class number coefficients completes the proof complex multiplication complex numbers composition compute conductor congruence Corollary cy² defined Dirichlet discriminant 4n elliptic curves Euler Fermat's finite fn(x form f(x,y form x² forms of discriminant formula fractional O-ideal Gal(L/K Gauss genus field genus theory given Hilbert class field Hint homomorphism ideal class group imaginary quadratic field implies integer solution isomorphism j-invariant ker(x lattice Legendre symbol Lemma minimal polynomial modular function nonzero norm number field number theory ny² odd prime p-function positive definite forms positive integer prime ideal prime to f proof of Theorem 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 Z/DZ