英语翻译The developments we have been describing thus far lasted

英语翻译
The developments we have been describing thus far lasted close to a century.They
gave rise to important “concrete” theories—Galois theory,algebraic number theory,
algebraic geometry—in which the (at times implicit) field concept played a central
role.
At the end of the nineteenth century abstraction and axiomatics were “in the air.”
For example,Pasch (1882) gave axioms for projective geometry,stressing for the
first time the importance of undefined notions,Cantor (1883) defined the real numbers as equivalence classes of Cauchy sequences of rationals,and Peano (1889) gave his axioms for the natural numbers.In algebra,von Dyck (1882) gave an abstract
definition of a group which encompassed both finite and infinite groups (about thirty
years earlier Cayley had defined a finite group),and Peano (1888) gave a definition
of a finite-dimensional vector space,though this was largely ignored by his contemporaries.The time was propitious for the abstract field concept to emerge.Emerge it did in 1893 in the hands ofWeber (of Dedekind–Weber fame).
Weber’s definition of a field appeared in his 1893 paper “General foundations of
Galois’ theory of equations” [23],in which he aimed to give an abstract formulation
of Galois theory:
In the following an attempt is made to present the Galois theory of algebraic equations in a way which will include equally well all cases in which this theory might be used.Thuswepresent it here as a direct consequence of the group concept illuminated by the field concept,as a formal structure completely without reference to any numerical interpretation of the elements used.
Weber’s presentation of Galois theory is indeed very close to the way the subject
is taught today.His definition of a field,preceded by that of a group,is as follows:
A group becomes a field if two types of composition are possible in it,the
first of which may be called addition,the second multiplication.The general
determination must be somewhat restricted,however.
1.We assume that both types of composition are commutative.
2.Addition shall generally satisfy the conditions which define a group.
Although the associative law under multiplication is missing,and the axioms are
not independent,they are of course very much in the modern spirit.As examples
of his newly-defined concept Weber included the number fields and function fields
of algebraic number theory and algebraic geometry,respectively,but also Galois’
finite fields and Kronecker’s “congruence fields” K[x]/(p(x)),K a field,p(x) a
polynomial irreducible over K.
Weber proved (often reproved,after Dedekind) various theorems about fields
which later became useful in Artin’s formulation of Galois theory,and which are
today recognized as basic results of the theory.
后面这个翻译不翻译都可以的。如果 磨练下自己，那我太感激了。
In an 1899 article entitled “New foundations of the theory of algebraic numbers,”
Hensel began a life-long study of p-adic numbers.Inspired by the work of Dedekind–
Weber we have described above,Hensel took as his point of departure the analogy
between function fields and number fields (p.55).Just as power series are useful for
a study of the former,Hensel introduced p-adic numbers to aid in the study of the
latter:
The analogy between the results of the theory of algebraic functions of one
variable and those of the theory of algebraic numbers suggested to me many
years ago the idea of replacing the decomposition of algebraic numbers,with
the help of ideal prime factors,by a more convenient procedure that fully
corresponds to the expansion of an algebraic function in power series in the
neighborhood of an arbitrary point.

发展到目前为止我们已经描述了近一个世纪.他们
引起混凝土”重要”theories-Galois理论、代数数论、
代数geometry-in(当时的隐式的)领域扮演重要的概念
的作用.
在到19世纪末,抽象和axiomatics "在空气”.
例如,Pasch(1882年)给公理,强调对射影几何的
第一次未定义的重要性概念,康托(1883年)定义实数与相等类柯西序列的rationals,Peano(1889年)给公理的自然数字.代数中,冯·代(1882年)把抽象的
一组定义包括两个有限与无限组(大约三十岁
年前定义了一个有Cayley有限群),Peano(1888)给一个定义
基于有限维向量空间的,尽管这是在很大程度上忽略了他同时代的人.当时领域更有利于抽象概念出现.1893年,出现了手中的ofWeber Dedekind-Weber名望).
韦伯的定义1893年出现在他领域的基础论文一般
Galois方程的理论”(23),他的目标是给了的现实的一种抽象的制定
Galois的理论与方法:
在接下来的尝试Galois理论提出的线性代数方程组的方法同样包括所有案件中,这个理论可以用Thuswepresent它在集团的一个直接后果领域照亮的概念形式结构理念,完全没有引用数值解释的元素使用.
韦伯陈述的Galois理论真的是非常接近目标
教育的今天.他的定义的领域,在一个集团,是如下:
一组领域成为两种成分如果,这是有可能的
第一本可以被称为.此外,第二乘法.一般
必须确定有所限制,但.
1.我们假设这两种类型的成分就都交换性.
2.一般满足条件还定义一组.
虽然结合律在多元化的丢失,和公理
不是相互独立的,他们当然非常现代精神.为例
他的newly-defined韦伯数概念包括田地和功能领域
代数数的理论和代数几何、分入账,但也Galois”
基于有限域和Kronecker的“同余场”凯西[x]/(p(x)),凯西方面,p(x)
多项式不可约在K.
韦伯后,证明(通常是责备各种定理Dedekind领域
后来的有用的Artin制定Galois理论,是吗
今天的结果确认为基本理论.
第二段
1899年的“新篇题为《理论的基础代数数字.”
Hensel开始的终身学习p-adic数字.工作的灵感Dedekind -
如上所述,我们与Hensel照他的起点比喻
功能领域之间和数量字段(p.55).正如幂级数是有用的
以前的研究,介绍了Hensel p-adic编号以助于研究
后者:
结果的相似性代数的理论之一.功能
变量和代数理论建议.我许多数字
年前取代的想法的线性代数数据、分解
理想的帮助下分解质因数,用更方便的程序,充分
与一种代数的扩展功能的幂级数
任意点附近.