Number Theory With Emphasis on Algorithms and Algebraic Number Theory

摘要亨德里克·伦斯特拉·伯克利9732709伦斯特拉教授将研究算法数论、代数论和代数几何中的一系列问题。所研究的课题包括光滑数、梅森素数、局部域的伽罗瓦群、Belyi函数、缺位多项式和有限域上的多项式。数论是数学中最古老的分支之一。它研究的是自然数，也就是计数的基数。从该地区一开始，人们就知道对所有数字都是可能的，而对于一个特定的数字，实际可以做的是什么，之间就存在着差距。例如，众所周知，任何自然数都只能以一种方式分解为较小的不可分解数的乘积。然而，考虑到精心选择的大量数字，将其分解成各个部分可能非常困难，以至于科学家们开发了基于因式分解的安全代码。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，改进的计算方法的潜在应用远远超出了数学和科学，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.