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1.
利用局部域的方法,研究素数p在有理数域的6次根扩张中的素理想分解问题,并完全确定了素数p在Q(6√u)中分解所可能有的形式(pα︱︱u).为进一步研究素数p在Q(2l√u)中(l为素数)的分解提供了途径. 相似文献
2.
In this paper, we describe many improvements to the number field sieve. Our main contribution consists of a new way to compute individual logarithms with the number field sieve without solving a very large linear system for each logarithm. We show that, with these improvements, the number field sieve outperforms the gaussian integer method in the hundred digit range. We also illustrate our results by successfully computing discrete logarithms with GNFS in a large prime field.
3.
《Discrete Mathematics》2024,347(1):113665
Recently, in coding theory and cryptography, it has been important the diversified use of lattices. One use of them is to cover a space. Each lattice has a covering radius, a number corresponding to the radius of a ball whose translations by all the points of the lattice cover efficiently the space generated by a basis of it. A way to obtain lattices algebraically is from subgroups of the multiplicative group of units of a number field via the logarithm embedding. This includes the logarithm lattice. In this work, it is presented an upper bound on the covering radius of the logarithm lattice obtained from the units of general cyclotomic number fields via the logarithm embedding, which generalizes an upper bound present in a previous work for cyclotomic number fields of prime-power indices. 相似文献
4.
《Journal of Complexity》2000,16(2):411-423
This paper provides verification procedures for a number of decision problems in quadratic function fields of odd characteristic, thereby establishing membership of these problems in both NP and co-NP. The problems include determining the ideal and divisor class numbers of the field, the regulator of the field (in the real case), a generating system of the ideal class group, a basis of the ideal class group, the pricipality of an ideal, the equivalence of two ideals, the discrete logarithm of an ideal class with respect to another ideal class, and the order of a class in the ideal class group. While several of these problems belong to the aforementioned complexity classes unconditionally, others require a certain assumption to ensure that the verification procedures can be done in polynomial time; so far, this assumption has only been verified for fields of high genus. 相似文献
5.
设K/F是整体函数域的素数l次循环扩张,F是有理函数域F_q(T)上的有限可分扩域.利用函数域的Conner-Hurrelbrink正合六边形与源于短正合列的正合六边形,本文在l整除与不整除基域F的理想类数的情形下,分别研究函数域K理想类群的Sylow l-子群的结构.同时,利用得到的结果,本文给出了基域F的单位为K中元素norm的若干条件. 相似文献
6.
We consider the well-known discrete logarithm problem in a finite simple field GF( $p$ ), where $p$ is a prime number, which has several application in problems of information protection. In Sec. 1, we introduce and study some number sequences arising in the continued fraction expansion of a real number. The results obtained are used in Sec. 2, where we introduce a new algorithm based on rational approximations for solving the problem of representing the discrete logarithm of a given number as the sum of logarithms of small primes; this problem is an important part of the discrete logarithm problem. We obtain several results necessary to construct and justify the representation algorithm. This algorithm is stated exactly in Sec. 3. We present several experimental results illustrating the work of the algorithm for prime numbers of the order of 10161031. 相似文献
7.
Elliot Benjamin 《The Ramanujan Journal》2006,11(1):103-110
Let
be an imaginary biquadratic number field with Clk,2, the 2-class group of k, isomorphic to Z/2Z × Z/2mZ, m > 1, with q a prime congruent to 3 mod 4 and d a square-free positive integer relatively prime to q. For a number of fields k of the above type we determine if the 2-class field tower of k has length greater than or equal to 2. To establish these results we utilize capitulation of ideal classes in the three unramified
quadratic extensions of k, ambiguous class number formulas, results concerning the fundamental units of real biquadratic number fields, and criteria
for imaginary quadratic number fields to have 2-class field tower length 1.
2000 Mathematics Subject Classification Primary—11R29 相似文献
8.
运用局部域理论给出了奇素数p在数域K=Q(u~(1/2),v~(1/2))上的素理想分解形式,其中l是奇素数,u,v∈z~*,且u/vQ~l. 相似文献
9.
Jürgen G. Hinz 《Monatshefte für Mathematik》1985,100(4):259-275
Generalizing a method introduced by Elliott in the rational case to number fields in an appropriate way, asymptotic estimates are given for the number of algebraic primes in certain parallelotopes which are primitive roots for almost all (in a certain sense) prime ideal moduli. The proofs depend upon a fundamental lemma of Selberg's rational sieve method and make use of the large sieve in the setting of an algebraic number field. 相似文献
10.
11.
L. F. Kondakova 《Mathematical Notes》1971,10(1):468-473
A variant of the large-sieve: method, using a combination of results obtained by Lavrik, Montgomery, and Eombieri, is employed to derive asymptotic properties of the number of solutions of the equationNp+Na=n wherep is a prime ideal of some ideal class of a field K of degree n4, anda is a prime ideal of a class of an imaginary quadratic field.Translated from Matematicheskie Zametki, Vol. 10, No. 1, pp. 73–81, July, 1971.The author wishes to thank A. I. Vinogradov for his help and advice in this work. 相似文献
12.
We give more efficient criteria to characterise prime ideal or primary ideal. Further, we obtain the necessary and sufficient conditions that an ideal is prime or primary in real field from the Gröbner bases directly. 相似文献
13.
I. A. Popovyan 《Mathematical Notes》2006,80(1-2):72-82
In the present paper, a polynomial algorithm is suggested for reducing the problem of taking the discrete logarithm in the ring of algebraic integers modulo a power of a prime ideal to a similar problem with the power equal to one. Explicit formulas are obtained; instead of the Fermat quotients, in the case of residues in the ring of rational integers, these formulas use other polynomially computable logarithmic functions, like the $\mathfrak{p}$ -adic logarithm. 相似文献
14.
Lesseni Sylla. 《Mathematics of Computation》2006,75(255):1519-1526
We prove that there is no primitive octic number field ramified only at one small prime, and so no such number field with a nonsolvable Galois group.
15.
Vahap Erdoğdu 《Archiv der Mathematik》2009,93(3):213-217
We call an ideal I of a commutative ring R radically perfect if among the ideals of R whose radical is equal to the radical of I the one with the least number of generators has this number of generators equal to the height of I. Let R be a Noetherian integral domain of Krull dimension one containing a field of characteristic zero. Then each prime ideal of
the polynomial ring R[X] is radically perfect if and only if R is a Dedekind domain with torsion ideal class group. We also show that over a finite dimensional Bézout domain R, the polynomial ring R[X] has the property that each prime ideal of it is radically perfect if and only if R is of dimension one and each prime ideal of R is the radical of a principal ideal. 相似文献
16.
本文对一类初等几何定理的证明给出了一种机械化方法,利用这种方法,可计算出一个由有限个素理想组成的集合,所有属于假设部分对应的某一扩域上的理想的素理想都在这个集合中出现并且可以挑选出来.因而一个几何定理一般真确,当且仅当终结多项式属于全部的这种素理想,即对其不可约特征列的余式为零. 相似文献
17.
In this paper, we propose algorithms for computing differential Chow forms for ordinary prime differential ideals which are given by characteristic sets. The algorithms are based on an optimal bound for the order of a prime differential ideal in terms of a characteristic set under an arbitrary ranking, which shows the Jacobi bound conjecture holds in this case. Apart from the order bound, we also give a degree bound for the differential Chow form. In addition, for a prime differential ideal given by a characteristic set under an orderly ranking, a much simpler algorithm is given to compute its differential Chow form. The computational complexity of the algorithms is single exponential in terms of the Jacobi number, the maximal degree of the differential polynomials in a characteristic set, and the number of variables. 相似文献
18.
Soogil Seo 《Journal of Number Theory》2005,115(2):348-359
In his paper (Invent. Math. 109 (1992) 329-350), Solomon finds an information on the prime factorization of an element coming from a circular unit 1-ζ over the ideal class group of a real abelian number field L, where ζ denotes a root of unity. Using this he obtains an annihilator of the p-Sylow subgroup of the subgroup of the ideal class group of L generated by the classes of prime ideals lying above p. We generalize this result to the circular distributions which has the axiomatic definition of Euler systems as its defining property. 相似文献
19.
20.
Noureen A. Khan 《Journal of Pure and Applied Algebra》2019,223(2):504-514
The geometric representation of a knot is not too dissimilar from a graph and this interaction has helped mathematicians to solve many problems. In this paper, we apply graph theory tools to study the classification of virtual knots and links. We define virtual planar graphs and compute virtual path width of an associated graph of a virtual link. We show that the virtual path width of an associated graph is equal to the virtual bridge number of a pseudo prime knot. 相似文献