Algebra
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Algebraic number theory
José Othon Dantas Lopes
Algebraic number theory originated from the attempts to determine integer and rational solutions of algebraic equations. The search for such solutions has pushed the advance of mathematics with the creation of new fields of study. There are many famous problems related to algebraic equations that have fascinated mathematics geniuses. For example, Euler became interested in Fermat’s several unproven statements. Euler’s efforts to prove Fermat’s conjectures led him naturally to the search of integers that are congruent to a square modulo a prime number. Nowadays those integers are known as quadratic residues (of the prime number under consideration). For a given prime p, the problem of deciding whether an integer n is a quadratic residue modulo p can be solved efficiently, that is, in polynomial time, via Euler’s criterion. A more interesting problem is to determine the primes of which a given integer is a quadratic residue. This problem led Euler to discover the Law of Quadratic Reciprocity from numerical evidence. However, he did not prove that his formula was correct. Legendre, who discovered the same law independently from Euler, published several proofs of his theorem, but all of them were flawed. Gauss, who discovered the law independently, provided its first correct proof when he was 17. Gauss wrote that “for an entire year this theorem tormented me and absorbed my greatest efforts until at last I obtained a proof.” Gauss also presented at least five more different proofs of the Law of Quadratic Reciprocity.
On the other hand, algebraic numbers, that is, complex numbers that are roots of polynomials with rational coefficients, come into play in a natural way as a tool to solve the aforementioned problems. A number field is a finite extension of the field of rational numbers (note that a number field is comprised of algebraic numbers). The elements of a number field that are roots of monic polynomials with integer coefficients are called algebraic integers and they are particularly interesting. It is a basic result in algebraic theory the set of algebraic integers of a number field forms a ring. The geometrical representation of integral ideals of this ring represents one of the strongest links between algebraic number theory and many other areas of science. Geometry of numbers is an important part of number theory. Its origins lie in the works of Minkowski published around 1896. The focus of the theory is on the conversion of arithmetic problems into geometry problems.
The geometrical representation of any ideal contained in the ring of integers of a number field of degree n is a lattice, that is, a free Z-module of rank n in the n-dimensional Euclidean space. The choice of ideals whose geometrical images are lattices of high density is one of the most intriguing and challenging problems in geometry of numbers, both from the theoretical and the practical viewpoints. On the pratical side, dense sphere packings can be used to solve the channel coding problem, that is, the design of signal sets for data transmission and storage. Our group has made progress in regards to producing dense lattice packings, especially those associated cyclotomic number fields.
Given a positive integer n, the problem of determining the number field of degree n of smallest discriminant in absolute value is a classical problem in number theory. We have solved particular cases of the problem, viz., those where the field is restricted to be Abelian.
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