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[已转移到维基条目] [NumberFields@home]项目研究介绍

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发表于 2013-7-3 02:51:20 | 显示全部楼层 |阅读模式

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辛苦啦~~: 5.0
辛苦啦~~: 5
  发表于 2013-7-3 19:21

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参与人数 1维基拼图 +8 收起 理由
昂宿星团人 + 8

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 楼主| 发表于 2013-7-3 02:53:30 | 显示全部楼层
本帖最后由 fwjmath 于 2013-7-3 04:04 编辑

NumberFields@home Project Description
NumberFields@home项目介绍

Project Goals

项目目的

Fields are important mathematical constructs that have far reaching applications to many branches of mathematics.  Many people are familiar with the fields of rational numbers, real numbers, and complex numbers.  The fields we are concerned with in this project are finite extension fields of the rational numbers.  In particular, we are interested in imprimitive degree 10 fields (called decic fields).  Computing lower degree fields requires less processing power and have already been sufficiently tabulated.  The degree 10 case is the first case requiring a massively parallel solution, and hence the reason for implementing a BOINC project.

域是一种重要的数学结构,它在数学的许多分支有着影响深远的应用。很多人对有理数域,实数域和复数域都颇为熟悉。在这个项目中,我们主要研究的是有理数域的有限扩张。更准确地说,是次数为十的域(或称十次数域)。扩张次数更低的域需要的计算资源更少,而我们已经有一张记录它们的表。十次数域是第一种需要高度并行计算的情况,而这正是这个BOINC项目存在的理由。

One way to categorize fields is by the primes that ramify in them.  For a given set of primes, the number of fields ramified at those primes is finite.  The primary goal of the project is to find this finite set of fields for various sets of primes. Since the number of combinations of primes is unlimited, the project will remain open-ended for the forseeable future.

对数域分类的一种方法是按照其中的分歧素数来分类。对于一个给定的素数集合,以这些素数为分歧素数的数域只有有限个。本项目的主要目的就是找到不同分歧素数集合对应的数域。由于素数集合永无止尽,在可以预见的将来,本项目一直会有工作。

Another way to categorize fields is by their discriminant, which is an important invariant for a field.  Given a fixed bound \(B\), there are only a finite number of fields whose discriminant is less than this bound.  A secondary goal of the project is to determine the finite set of "minimum discriminant" imprimitive decic fields for the bound \( B=1.2 \times 10^{11} \).  We chose this bound for it's potential to find more fields while keeping the computational load manageable.

另一种分类的方法是根据它们的判别式分类。判别式是数域中非常重要的一个不变量。给定一个上界B,判别式小于这个上界的数域是有限的。本项目的次要目的就是确定判别式小于B=1.2*10^11的非本原十次数域的完整列表。我们选择这个上界是为了平衡能找到的数域个数与所需计算量。

Applications of the Project

项目应用

Cryptography

密码学

Number fields have applications to many parts of number theory, including parts that have direct application to cryptography.  For example, number fields are used in some modern factoring algorithms which are relevant to RSA.

数域在数论的方方面面都有用途,其中包括在密码学中的直接应用。例如,在与RSA密码算法有关的大数分解算法中就用到了数域(译者注:应该是指数域筛法,number field sieve)。

Automorphic Forms

自守形式

The theory of automorphic formsis an important topic within mathematics.  Automorphic forms generalize the concept of modular forms to functions of several complex variables. There are deep connections between automorphic forms and finite field extensions with special ramification properties.  For example, Calegari was studying automorphic forms over \( \bbQ(i) \) and needed to show that there were no automorphic forms of a certain type.  The problem was recast into a statement about number fields, summarized by the following conjecture:

自守形式理论在数学中是个重要的题目。它将模形式从单变量复函数推广到多变量的情况。自守形式与拥有某些特殊分歧性质的有限域扩张有着深刻的联系。比如说,Calegari在研究Q(i)上的自守形式时,希望能证明不存在某种特定的自守形式。这个问题可以转化为关于数域的问题,给出以下的猜想:

Calegari's Conjecture:

Calegari猜想

There is no quintic extension \( L \) of \( \bbQ(i) \) satisfying
  •   \( \Gal( L^g/\bbQ(i))=A_5 \) (where \(L^g\) is the Galois closure of \(L\) ),
  •   \( L \) is unramified outside of \( S=\{2,5\} \), and
  •   \( d_L \) divides \( 2^{14}5^{15} \).

不存在Q(i)的五次域扩张L,使得:
  • L的伽罗华闭包与Q(i)的商的伽罗华群是交错群A_5
  • L的分歧素数不包含2和5以外的素数
  • L的判别式整除2^14*5^15

Using the table of decic fields over \( \bbQ(i) \), Calegari's conjecture was shown to be true.  (Note that the project is still completing the search over \( \bbQ(i) \), but the discriminant bound in Calegari's conjecture was small enough that the earlier part of the search was sufficient to determine the truth of the conjecture.)

利用Q(i)的十次数域表,我们证明Calegari猜想是正确的。(备注:虽然项目仍然在搜索Q(i)上的十次数域,但Calegari猜想中对判别式的界足够小,之前的搜索足以确定它的正确性)

Galois Theory

伽罗华理论

The fields obtained by this project can be further categorized by their Galois group. With large enough tables, one can make conjectures relating to distributions of the various Galois groups.  A Galois root discriminant (GRD) can also be computed for each field in the tables, which may have importance to those interested in low GRD fields.

项目搜索到的数域可以进一步按照它们的伽罗华群分类。如果有足够大的数域列表,我们能对不同伽罗华群的分布进行猜想。对于列表中的每个数域,我们也可以计算它们的伽罗华根判别式(GRD),这对于研究低GRD数域的数学家很有用处。

Theoretical Physics

理论物理学

The fields concerned with in this project have connections to the p-adic fields. In recent years, p-adic analysis has been applied to problems in theoretical physics, including quantum mechanics and string theory.  Hereis a good introduction to the relevant concepts.  It is too early to tell exactly how beneficial our tables of fields will be to the physics community; we are basically charting unknown territory here.

本项目研究的数域与p进域有联系。在最近,p进分析被应用到理论物理的一些问题上,比如说量子力学与弦论。这里有一个对相关概念的介绍。现在要说我们的列表对于物理学家来说有什么用处,仍然为时尚早;我们仍然在探索未知的领域。

Algorithmic Details

算法细节

Finite extension fields are represented by polynomials (i.e. they are of the form \( \bbQ(\alpha) \) where \(\alpha\) is the root of a polynomial).  Bounds on the field discriminant give rise to bounds on the polynomial coefficients, so there are a finite number of possible polynomials that can represent the fields we are searching for.  At the most basic level, the algorithm searches over this finite set of polynomials, checking whether or not a polynomial can represent a field with the desired discriminant and ramification properties.  At a finer level, the algorithm uses some tricky theoretical arguments to reduce the polynomialsearch space.  In addition, the targetted ramification structure gives rise to congruence relations on the polynomial coefficients, which further reduces the search space.  Anybody interested in the finer details of the algorithm is encouraged to look through my dissertation.  

域的有限扩张可以用多项式表示,它们可以表示为Q(a),其中a是某个多项式的根。域的判别式能给出多项式系数的界,所以我们的搜索只需要处理有限个多项式。粗略地说,我们的算法遍历所有可能的多项式,检查它们是否表示了一个符合我们要求的数域。更进一步地说,我们的算法用了一些复杂的数学论证来减小搜索空间。另外,指定的分歧结构也会给出多项式系数的一些同余关系,这也可以减小搜索空间。对算法有兴趣的人可以查看我的博士论文。

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参与人数 1维基拼图 +15 收起 理由
昂宿星团人 + 15 校对不能。。

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发表于 2013-7-22 23:19:19 | 显示全部楼层
校对不能+1,读起来也挺正常的,搬运了算了…………………………………………
膜拜楼上
发表于 2013-7-24 04:34:02 | 显示全部楼层
本帖最后由 arthur200000 于 2013-7-24 04:38 编辑
(i.e. they are of the form \( \bbQ(\alpha) \) where \(\alpha\) is the root of a polynomial)
它们可以表示为Q(a),其中a是某个多项式的根

@fwjmath 这边是奇怪的TEX标记? 如果是的话应该改成α对吧→ →
唔这边的TEX太多了……作者疑似LATEX的粉丝

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很精神嘛今天?  发表于 2013-7-24 06:17

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参与人数 1维基拼图 +3 收起 理由
昂宿星团人 + 3 从字面上看,我也赞同,原文写的是alpha嘛.

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 楼主| 发表于 2013-7-24 07:20:48 | 显示全部楼层
arthur200000 发表于 2013-7-24 04:34
@fwjmath 这边是奇怪的TX标记? 如果是的话应该改成α对吧→ →
唔这边的TX太多了……作者疑似LTX的粉 ...

对的,这个是tex的标记。其实用a和用alpha也差不多,在书上两种记号都有人用,因为简便我就用a了,当然改成alpha也可以。
发表于 2013-7-24 09:19:18 | 显示全部楼层
@昂宿星团人 一我目前是时差党,跟着米帝的EST……
二我刚发过烧……

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参与人数 1基本分 +20 收起 理由
昂宿星团人 + 20 抚恤金发放

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发表于 2013-7-28 05:05:35 | 显示全部楼层
Copied to wiki.

btw,"invarient"似乎可以译为"常量"......

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数学上一般是这样翻译的……拓扑不变量之类的……  发表于 2013-7-28 07:45

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参与人数 1维基拼图 +3 收起 理由
昂宿星团人 + 3 最近没时间看,总之先加分。。.

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