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quadratic fields
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  二次域
     An upper bound for class numbers of real quadratic fields Q(p~(1/2)) with p≡3(mod4)
     实二次域Q(p~(1/2))(p≡3(mod4))类数的上界
短句来源
     Positive definite unimodular lattices over the real quadratic fields Q(8k+1)~(1/2)
     实二次域Q(8k+1)~(1/2)上的正定幺模格
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     The Divisibility of Class Numbers of Imaginary Quadratic Fields Q(a~2-4k~n)
     虚二次域Q(a~2-4k~n)~(1/2)类数的可除性
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     ON THE CONDITIONS OF INTEGERS βWHICH HAEV PRIMITIVE ROOTS IN THE QUADRATIC FIELDS K(±2)~(1/2)
     论二次域 K((±2)~(1/2))内整数β有原根存在的条件
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     An Upper Bound for Class Numbers of Real Quadratic Fields Q(■) with p≡3(mod 4)
     实二次域Q(P(1/2))(p≡3(mod 4))类数的上界
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  “quadratic fields”译为未确定词的双语例句
     ON x~2 +y~2=m IN THE QUADRATIC FIELDS
     关于不定方程x~2+y~2=m
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     Narrow Class Groups of Quadratic Fields with Three Odd Prime Divisors in Discriminants
     具有三个奇素因子的判别式的二次数域的狭义类群
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     In this paper, the sufficient conditions that there's no present of Euclidean algorithm in some quadratic fields are given.
     给出了某些二次数域中不存在Euclid算法的一个充分条件.
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  相似匹配句对
     A Congruence on Real Quadratic Fields
     实二次域的一个同余式(英文)
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     A Result of Real Quadratic Function Fields
     实二次函数域的一个结果
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     Unification of fields
     场的统一
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     These fields are of the H.R.
     场具有H . R .
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     The Quadratic Exponential Generator
     二次指数发生器
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  quadratic fields
We shall show that for any ? >amp;gt; 0, the number of imaginary quadratic fields with discriminant ≥ - X and ideal class group having a subgroup isomorphic to ?/g ?× ?/g ? is ? X1/g-?.
      
Imaginary quadratic fields with noncyclic ideal class groups
      
Distribution of prime numbers in two imaginary quadratic fields.
      
Scalar products of Hecke L-series of quadratic fields
      
We suggest an infinite family of such cases covering all the imaginary quadratic fields.
      
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Let K denote a field of type ■,i.e.an extension of degree 2~nover Q which is the composite of n distinct quadratic fields.In this paper,westudy this class of fields for general n:(1)Classifying all fields of this type and proving that there are exactly4(6 resp.)possible cases of the decomposition of an odd prime p(2 resp.)in K.(2)Finding all the prime ideals of K explicitly.(3)Determining an integral basis and the discriminant of K.(4)Proving that there are only 6 possible groups(ζ(?))of roots...

Let K denote a field of type ■,i.e.an extension of degree 2~nover Q which is the composite of n distinct quadratic fields.In this paper,westudy this class of fields for general n:(1)Classifying all fields of this type and proving that there are exactly4(6 resp.)possible cases of the decomposition of an odd prime p(2 resp.)in K.(2)Finding all the prime ideals of K explicitly.(3)Determining an integral basis and the discriminant of K.(4)Proving that there are only 6 possible groups(ζ(?))of roots of unityin K(ω=2,4,8,6,12,24),and obtaining a formula of class number of K.

■型数域K,即是由n个二次域合成的Q的2~n次扩城.本文对一般的n,较系统地研究了这类域的各代数结构.§2给出了域K的分类和标准生成,证明了奇素数p(以及2)在K 恰有4(以及6)种可能素分解情形.§3明显构作出K 的诸素理想.§4给出K 的整基、判别式和导子.§5证明了K 仅可能含2、4、6、8、12、24次单位根;从§2的结果导出K 的类数公式.上述诸结果与二次域的相应结果极为相似,是后者的自然拓广.

In 187S, E.Lucas asked whether the Diophantine equationhas only nontrivial solution x=24, y=70. This was solved (affirmatively) by Watson, using elliptic functions, and by Ljunggren, using a Pell equation in a quadratic field. Mordell asked if there was an elementary proof. In this paper, we have given fcn elementary proof of, equation (1). hence we have solved the Mordll's question,

1975年,E.Lucas问丢番图方程:6y~2=x(x+1)(2x+1) (1)是否仅有非平凡解r=24,y=70。Watson和Ljunggren给出了肯定的回答。他们分别利用椭圆函数和四次扩域上的Pell方程给出了证明,证明很复杂.Mordell问是否有一个初等证明.本文给出了一个完全初等的证明,因而解决了Mordell提出的问题。

By making use of radical extensions over cyclotomic fields, we try to depict definition polynomials of cyclic extensions over Q of prime degree p and generalize the well-known correspondence between all quadratic fields and subgroups id Q/Q2 of order 2.Although the polynomials look somewhat "clumsy" since they involve complex combinatoral equations, the results in the cubic case which coincide with Prof. Li's are truly beautiful; and when p=5, we find the interestingTheorem. The splitting field is...

By making use of radical extensions over cyclotomic fields, we try to depict definition polynomials of cyclic extensions over Q of prime degree p and generalize the well-known correspondence between all quadratic fields and subgroups id Q/Q2 of order 2.Although the polynomials look somewhat "clumsy" since they involve complex combinatoral equations, the results in the cubic case which coincide with Prof. Li's are truly beautiful; and when p=5, we find the interestingTheorem. The splitting field is wildly ramified quintic cyclic over Q for all aeZ, where (x) is the quintie cyclotomic polynomial.

利用分圆域Q(ζ)的根式扩张,对有理数域的(奇素数)p次循环扩张进行了研究,得到了任意p次循环扩张的定义方程.文[1],[2]在p=3时已有了很好的结果;本文讨论了p=5的情形,最后得到一类由含一个有理参数的五次方程确定的五次循环扩张.

 
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