On imaginary quadratic function fields with the ideal class group to be exponent ≤2

A necessary condition is presented for the ideal class group of an imaginary quadratic function field K=k(D) (k=F-q(x), 2q) to be of exponent ≤2. The condition is proved to be sufficient in some cases. An analogue of Louboutin’s result in function field case is particularly presented.

Subgroups of ideal class groups of real quadratic algebraic function fields

Necessary and sufficient condition on real quadratic algebraic function fields K is given for theirideal class groups H(K) to contain cyclic subgroups of order n. And eight series of such real quadratic functionfields K are obtained whose ideal class groups contain cyclic subgroups of order n. In particular, the ideal classnumbers of these function fields are divisible by n.

THE DETERMINATION OF SUBGROUPS IN IDEAL CLASS GROUPS OF REAL QUADRATIC FIELDS

Series of results about ideal class groups H （m） and class numbers h （m） of real quadratic fields Q (m1/2) can be obtained from [6]. Some of them will be shown in this note. We denote by Cn =Z/nZ the cyclic group of order n. Let m ∈ Z denote a square free positive integer, and let z1, z, t ∈Z be arbitrary integers with z1 odd and t】0.

Infinite Family of Imaginary Cyclic Fields of Degree p-1 with the p-Rank of the Ideal Class Groups of at Least Two

Imaginary cyclic fields of degree p-1 which have two distinct unramified cyclic extensions of degree p are produced using elementary properties of the Lucas sequences. An infinite family of imaginary cyclic fields of degree p-1 are then given with the p-rank of the ideal class groups of at least two.

A kind of signature scheme based on class groups of quadratic fields

Quadratic-field cryptosystem is a cryptosystem built from discrete logarithm problem in ideal class groups of quadratic fields(CL-DLP). The problem on digital signature scheme based on ideal class groups of quadratic fields remained open, because of the difficulty of computing class numbers of quadratic fields. In this paper, according to our researches on quadratic fields, we construct the first digital signature scheme in ideal class groups of quadratic fields, using q as modulus, which denotes the prime divisors of ideal class numbers of quadratic fields. Security of the new signature scheme is based fully on CL-DLP. This paper also investigates realization of the scheme, and proposes the concrete technique. In addition, the technique introduced in the paper can be utilized to realize signature schemes of other kinds.

Ideal Class Groups and Subgroups of Real Quadratic Function Fields

In this paper, we study the real quadratic function fields K=k(D), given a necessary and sufficient condition for the ideal class group H(K) of any real quadratic function field K to have a cyclic subgroup of order n, and obtained eight series of such fields. The ideal class numbers h(O K) of K in the series all have a factor n.[

Ideal class groups and their subgroups of real quadratic fields

A necessary and sufficient condition is given for the ideal class group H( m } of a real quadratic field Q (m)to contain a cyclic subgroup of order n.Some criteria satisfying the condition are also obtained.And eight types of such fields are proved to have this property,e.g.fields with m=(zn+t+12)+4t(with t|zn-1),which contains the well-known fields with m=4zn+1 and m=z2n+4 as special cases.

Bounds of the Ideal Class Numbers of Real Quadratic Function Fields

The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields over k = F q (T). For five series of real quadratic function fields K, the bounds of h(D) are given more explicitly, e. g., if D = F 2 + c, then h(D) ≥ degF/degP; if D = (SG)2 + cS, then h(D) ≥ degS/degP; if D = (A m + a)2 + A, then h(D) ≥ degA/degP, where P is an irreducible polynomial splitting in K, c ∈ F q . In addition, three types of quadratic function fields K are found to have ideal class numbers bigger than one.

Lower Bound for Ideal Class Numbers of Real Quadratic Function Fields

In this paper, the theory of continued fractions of algebraic functions will be used to give a general theorem on lower bounds for class numbers of real quadratic function fields K=k(D). The bounds are given more explicitly for six types of real quadratic function fields. As a consequence, six classes of real quadratic function fields with ideal class number greater than one are given.[

SUBGROUPS OF CLASS GROUPS OF ALGEBRAIC QUADRATIC FUNCTION FIELDS

Ideal class groups H(K) of algebraic quadratic function fields K are studied. Necessaryand sufficient condition is given for the class group H(K) to contain a cyclic subgroup of anyorder n, which holds true for both real and imaginary fields K. Then several series of functionfields K, including real, inertia imaginary, and ramified imaginary quadratic function fields, aregiven, for which the class groups H(K) are proved to contain cyclic subgroups of order n.

关于丢番图方程D_1x^2+2~mD_2=y^n的解数

设Ｄ１、Ｄ２、ｍ、ｘ、ｙ是适合Ｄ１＞１，Ｄ２＞１，２Ｄ１Ｄ２，ｇｃｄ（Ｄ１，Ｄ２）＝ｇｃｄ（ｘ，ｙ）＝１的正整数，ｎ是适合ｎｈ的奇素数，其中ｈ是虚二次域Ｑ（－２ｍＤ１Ｄ２）的类数．本文主要证明了：方程Ｄ１ｘ２＋２ｍＤ２＝ｙｎ至多有５·１０１６组例外解（Ｄ１，Ｄ２，ｘ，ｙ，ｍ，ｎ），而且这些解都满足７ｎ＜８．５·１０６以及ｙｎ＜ｅｘｐ（ｅｘｐ（ｅｘｐ４６））．

Kummer函数域的理想类群(英文)

得到了二次函数域的理想类群的２－秩公式和一般ｌ次Ｋｕｍｍｅｒ函数域理想类群ｌ－秩的下界．由此完全确定出类数不被ｌ整除的所有ｌ次Ｋｕｍｍｅｒ函数域．其次，还证明了不存在这样的ｌ（ｌ＞２）次Ｋｕｍｍｅｒ函数域。

虚二次域的L-函数在中心点的值

给出了虚二次数域的一类L-函数在中心点的值,这为研究Dirichlet L-函数在中心点的算术意义提供了一条新的途径。

实二次域上理想类的zeta-函数在-1处的值

用连分数给出了实二次域理想类的zeta-函数-1处值的一个具体的计算公式.

实二次数域的一类L-函数在中心点的值(英文)

为了证明L(1/2,χ)≠0,本文定义了实二次域的一类L-函数并给出其在在中心点1/2处的值的表达式.

一类有关虚二次域理想类群的指数丢番图方程

设Ｄ１、Ｄ２是适合Ｄ１≥１，Ｄ２＞１，ｇｃｄ（Ｄ１，Ｄ２）＝１的无平方因子正整数，ＩＫ是虚二次域Ｋ＝Ｑ（－Ｄ１Ｄ２）的理想类群。本文证明了：如果存在正整数ａ、ｂ、ｋ以及奇素数ｐ，可使Ｄ１ａ２＋Ｄ２ｂ２＝ｋｐ，ｇｃｄ（ａ，ｂ）＝１，ｂ∈Ｎｐ，２ｋ，其中Ｎｐ是所有不含２ｔｐ±１之形素因数的正整数的集合，则当ｐ＞５且ｍａｘ（Ｄ１，Ｄ２）≥１０１０１９７时。

理想类群指数≤2的虚二次函数域

本文给出虚二次函数域K=k^（1/2）D（k=Fq（x）,2+q）的理想类群指数≤2的一个必要条件.在某些条件下,这个条件也是充分的.特别地,给出了Louboutin[1]一个结果在函数域情形的模拟.

8-rank of the class group and isotropy index

Suppose F = Q(√-p1 pt) is an imaginary quadratic number field with distinct primes p1,..., pt,where pi≡ 1(mod 4)(i = 1,..., t- 1) and pt ≡ 3(mod 4). We express the possible values of the 8-rank r8 of the class group of F in terms of a quadratic form Q over F2 which is defined by quartic symbols. In particular,we show that r8 is bounded by the isotropy index of Q.

New results on non-existence of generalized bent functions (Ⅱ)

Several new results on non-existence of generalized bent functions are presented by using the class group of related imaginary abelian number fields.

实二次域上理想类的Zeta-函数在-3处的值

用简单连分数给出了实二次域理想类的Zeta-函数在-3处值的一个具体的计算公式.