Number Theory With Emphasis on Algorithms and Algebraic Number Theory

Hendrik Lenstra伯克利9732709教授将研究算法数论、代数数论和代数几何方面的一系列问题。要研究的主题包括光滑数、梅森素数、局部域的伽罗瓦群、贝利函数、缺域多项式和有限域上的多项式。数论是数学最古老的分支之一。它是对自然数的研究，是计数的基本数字。从一开始，在已知所有数字的可能性和实际可以为特定数字做的事情之间就存在着差距。例如，我们知道任何自然数都只能被一种方法分解为更小的不可分解数的乘积。然而，考虑到一个精心挑选的大数字，它可能很难分解成部分，科学家们已经开发出基于分解的安全代码。这个项目致力于改进我们实际计算特定数字的一些最重要的数学性质的方法。在计算机时代，改进计算方法的潜在应用远远超出了数学和科学领域，进入了日常生活。

ABSTRACT Hendrik Lenstra Berkeley 9732709 Professor Lenstra will study a collection of problems in algorithmetic number theory, algebraic number theory, and algebraic geometry. Topics to be studied include smooth numbers, Mersenne primes, Galois groups of local fields, Belyi functions, Lacunary polynomials, and polynomials over finite fields. Number theory is one of the oldest branches of mathematics. It is the study of the natural numbers, the basic numbers of counting. From the very beginning of the area, there has been a gap between what was known to be possible for all numbers, and what could actually be done for a specific one. For example, it is known that any natural number can be factored in only one way as the product of smaller unfactorable numbers. Yet given a carefully chosen large number, it can be so difficult to factor into parts that scientists have developed secure codes based on factorization. This project is devoted to improving the methods we have for actually computing some of the most important mathematical properties of particular numbers. In the computer age, the potential applications for improved methods of computation go well beyond mathematics and science and into everyday life.