Hilbert's twelfth problem

About: Hilbert's twelfth problem is a research topic. Over the lifetime, 384 publications have been published within this topic receiving 9702 citations.

Number Theory in Function Fields

Abstract: Polynomials over Finite Fields.- Primes, Arithmetic Functions, and the Zeta Function.- The Reciprocity Law.- Dirichlet L-series and Primes in an Arithmetic Progression.- Algebraic Function Fields and Global Function Fields.- Weil Differentials and the Canonical Class.- Extensions of Function Fields, Riemann-Hurwitz, and the ABC Theorem.- Constant Field Extensions.- Galois Extensions - Artin and Hecke L- functions.- Artin's Primitive Root Conjecture.- The Behavior of the Class Group in Constant Field Extensions.- Cyclotomic Function Fields.- Drinfeld Modules, An Introduction.- S-Units, S-Class Group, and the Corresponding L-functions.- The Brumer-Stark Conjecture.- Class Number Formulas in Quadratic and Cyclotomic Function Fields.- Average Value Theorems in Function Fields.

Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex Multiplication

Abstract: FROM FERMAT TO GAUSS. Fermat, Euler and Quadratic Reciprocity. Lagrange, Legendre and Quadratic Forms. Gauss, Composition and Genera. Cubic and Biquadratic Reciprocity. CLASS FIELD THEORY. The Hilbert Class Field and p = x 2 + ny 2 . The Hilbert Class Field and Genus Theory. Orders in Imaginary Quadratic Fields. Class Fields Theory and the Cebotarev Density Theorem. Ring Class Field and p = x 2 + ny 2 . COMPLEX MULTIPLICATION. Elliptic Functions and Complex Multiplication. Modular Functions and Ring Class Fields. Modular Functions and Singular j--Invariants. The Class Equation. Ellpitic Curves. References. Index.