ON THE EXCEPTIONAL FIELDS FOR A CLASS OF REAL QUADRATIC FIELDS

In this paper, we give a lower bound exp（2.2 × 10~8 ） for those discriminants of real quadratic fields Q（√ d） with d= N^2-4 and h（d）=1.

New designing of cryptosystems based on quadratic fields

This paper proposes a method to construct new kind of non-maximal imaginary quadratic order （NIQO＊） by combining the technique of Diophantine equation and the characters of non-maximal imaginary quadratic order. It is proved that in the class group of this new kind of NIQO＊, it is very easy to design provable secure cryptosystems based on quadratic field （QF）. With the purpose to prove that this new kind of QF-based cryptosystems are easy to implement, two concrete schemes are presented, i.e., a Schnorr-like signature and an EIGamel-like encryption, by using the proposed NIQO＊. In the random oracle model, it is proved that： （1） under the assumption that the discrete logarithm problem over class groups （CL-DLP） of this new kind of NIQO＊ is intractable, the proposed signature scheme is secure against adaptive chosen-message attacks, i.e., achieving UF-CMA security; （2） under the assumption that the decisional Diffie-Hellman problem over class groups （CL-DDH） of this new kind of NIQO＊ is intractable, the enhanced encryption in this paper is secure against adaptive chosen-ciphertext attacks, i.e., reaching IND-CCA2 security.

On abelian 2-ramification torsion modules of quadratic fields

For a number field F and a prime number p, the Z_(p)-torsion module of the Galois group of the maximal abelian pro-p extension of F unramified outside p over F, denoted by T_(p)(F), is an important subject in the abelian p-ramification theory. In this paper, we study the group T_(2)(F) = T_(2)(m) of the quadratic field F = Q(m_(1/2)). Firstly, assuming m > 0, we prove an explicit 4-rank formula for quadratic fields that rk4(T_(2)(-m))= rk2(T_(2)(-m))-rank(R), where R is a certain explicitly described Rédei matrix over F_(2). Furthermore, using this formula, we obtain the 4-rank density formula of T_(2)-groups of imaginary quadratic fields. Secondly, for l an odd prime, we obtain the results about the 2-power divisibility of orders of T_(2)(±l) and T_(2)(±2l), both of which are cyclic 2-groups. In particular, we find that #T_(2)(l) ≡ 2#T_(2)(2l) ≡ h_(2)(-2l)(mod 16) if l ≡ 7(mod 8),where h_(2)(-2l) is the 2-class number of Q((-2l)_(1/2)). We then obtain the density results for T_(2)(±l) and T_(2)(±2l)when the orders are small. Finally, based on our density results and numerical data, we propose distribution conjectures about T_(p)(F) when F varies over real or imaginary quadratic fields for any prime p, and about T_(2)(±l)and T_(2)(±2l) when l varies, in the spirit of Cohen-Lenstra heuristics. Our conjecture in the T_(2)(l) case is closely connected to Shanks-Sime-Washington’s speculation on the distributions of the zeros of 2-adic L-functions and to the distributions of the fundamental units.

Some Results Connected with the Class Number Problem in Real Quadratic Fields

We investigate arithmetic properties of certain subsets of square-free positive integers and obtain in this way some results concerning the class number h（d） of the real quadratic field Q（√d）. In particular, we give a new proof of the result of Hasse, asserting that in this case h（d） = 1 is possible only if d is of the form p, 2q or qr. where p.q. r are primes and q≡r≡3（mod 4）.

A Note on 3-Divisibility of Class Number of Quadratic Field

In this paper,the authors show that there exists infinitely many family of pairs of quadratic fields Q(√D)and Q((√D+n)(1/2))with D,n∈Z whose class numbers are both divisible by 3.

An Additive Function on a Ring of Integers in the Imaginary Quadratic Field Q(d^(1/2))with Class-Number One

Let Bα（α）be an additive function on a ring of integers in the quadratic number field Q（（1/2）d）given by Bα（α）=∑p丨α*Nα（p）with a fixed α】0,where the asterisk means that the summation is over the non-associate prime divisors p of an integer α in Q（（1/2）d）,N（α）is the norm of α.In this paper we obtain the asymptotic formula of ∑N（α）≤x *Bα（α）in the case where the class-number of Q（（1/2）d）is one.

The Problem on Class Numbers of Quadratic Number Fields

It is a survey of the problem on class numbers of quadratic number fields.

p-Capitulation over Number Fields with p-Class Rank Two

Theoretical foundations of a new algorithm for determining the p-capitulation type ù(K) of a number field K with p-class rank ?=2 are presented. Since ù(K) alone is insufficient for identifying the second p-class group G=Gal(Fp2K∣K) of K, complementary techniques are deve- loped for finding the nilpotency class and coclass of . An implementation of the complete algorithm in the computational algebra system Magma is employed for calculating the Artin pattern AP(K)=(τ (K),ù(K)) of all 34631 real quadratic fields K=Q(√d) with discriminants 0d<108 and 3-class group of type (3, 3). The results admit extensive statistics of the second 3-class groups G=Gal(F32K∣K) and the 3-class field tower groups G=Gal(F3∞K∣K).

An Infinite Family of Number Fields with No Inert Primes

The goal of this paper is to show that there are infinitely many number fields K/Q, for which there is no inert prime p ∈ N*, i.e. ∀p ∈ N* a prime number, prime ideal of K such that where: Zk is the Dedekind domain of the integer elements of K. To prove such a result, consider for any prime p, the decomposition into a product of prime ideals of Zk, of the ideal . From this point, we use on the one hand: 1) The well- known property that says: If , then the ideal pZk decomposes into a product of prime ideals of Zk as following: . (where:;is the irreducible polynomial of θ, and, is its reduction modulo p, which leads to a product of irreducible polynomials in Fp[X]). It is clear that because if is reducible in Fp[X], then consequently p is not inert. Now, we prove the existence of such p, by proving explicit such p as follows. So we use on the other hand: 2) this property that we prove, and which is: If , is an irreducible normalized integer polynomial, whose splitting field is , then for any prime number p ∈ N: is always a reducible polynomial. 3) Consequently, and this closes our proof: let’s consider the set (whose cardinality is infinite) of monogenic biquadratic number fields: . Then each fθ(X) checks the above properties, this means that for family M, all its fields, do not admit any inert prime numbers p ∈ N. 2020-Mathematics Subject Classification (MSC2020) 11A41 - 11A51 - 11D25 - 11R04 - 11R09 - 11R11 - 11R16 - 11R32 - 11T06 - 12E05 - 12F05 -12F10 -13A05-13A15 - 13B02 - 13B05 - 13B10 - 13B25 -13F05

Quadratic Number Fields with Class Numbers Divisible by a Prime q

Let q 5 be a prime number. Let k d=() be a quadratic number field, where d =--gqq(1)2(1) ---qqqquwuq12((1)+). Then the class number of k is divisible by q for certain integers u,w. Conversely, assume W / k is an unramified cyclic extension of degree q (which implies the class number of k is divisible by q), and W is the splitting field of some irreducible trinomial f(X) = XqaXb with integer coefficients, k Df=(())with D(f) the discriminant of f(X). Then f(X) must be of the form f(X) = Xquq2wXuq1 in a cer-tain sense where u,w are certain integers. Therefore, k d=() with d =-----qqqqqquwuq(1)122(1)((1)+). Moreover, the above two results are both generalized for certain kinds of general polynomials.

Liouvillian and Analytic Integrability of the Quadratic Vector Fields Having an Invariant Ellipse

We characterize the Liouvillian and analytic integrability of the quadratic polynomial vector fields in R2 having an invariant ellipse.More precisely,a quadratic system having an invariant ellipse can be written into the form x=x2＋y2-1＋y（ax＋by＋c）,y=x（ax＋by＋c）,and the ellipse becomes x2＋y2=1.We prove that（i） this quadratic system is analytic integrable if and only if a=0;（ii） if x2＋y2=1 is a periodic orbit,then this quadratic system is Liouvillian integrable if and only if x2＋y2=1 is not a limit cycle;and（iii） if x2＋y2=1 is not a periodic orbit,then this quadratic system is Liouvilian integrable if and only if a=0.

A RANDOM TRANSPORT-DIFFUSION EQUATION

In this paper we study the integral curve in a random vector field perturbed by white noise. It is related to a stochastic transport-diffusion equation. Under some conditions on the covariance function of the vector field, the solution of this stochastic partial differential equation is proved to have moments. The exact p-th moment is represented through integrals with respect to Brownian motions. The basic tool is Girsanov formula.

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.[

A Remark on Computing the Tame Kernel of Quadratic Imaginary Fields

In this paper, we discuss a method to compute the tame kernel of a number field. Confining ourselves to an imaginary quadratic field, we prove that _υ:K_2~S F/K_2~S F→k~* is bijective when N_υ>8δ_D^6.

Computation of K_2 for the ring of integers of quadratic imaginary fields

This paper presents a method to get improved bounds for norms of exceptional v ’ s in computing the group K2 0F, where F is a quadratic imaginary field, and as an application we show that $K_2 \left[ {\left( {1 + \sqrt { - 43} } \right)/2} \right] = 1$

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.[

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.

Parametrization of the Quadratic Function Fields Whose Divisor Class Numbers are Divisible by Three

A parametrization of quadratic function fields whose divisor class numbers are divisible by 3 is obtained by using free parameters when the characteristics of the fields are not 3.

Unit groups of quotient rings of complex quadratic rings

For a square-free integer d other than 0 and 1, let K = Q（√d）, where Q is the set of rational numbers. Then K is called a quadratic field and it has degree 2 over Q. For several quadratic fields K = Q（√d）, the ring Rd of integers of K is not a unique-factorization domain. For d 〈 0, there exist only a finite number of complex quadratic fields, whose ring Rd of integers, called complex quadratic ring, is a unique-factorization domain, i.e., d = -1,-2,-3,-7,-11,-19,-43,-67,-163. Let Q denote a prime element of Rd, and let n be an arbitrary positive integer. The unit groups of Rd/（Q^n） was determined by Cross in 1983 for the case d = -1. This paper completely determined the unit groups of Rd/（Q^n） for the cases d = -2, -3.

On Quasi-Reduced Quadratic Forms

With the help of continued fractions, we plan to list all the elements of the set Q△ = {aX2 ＋ bXY ＋ cY2 ： a,b, c ∈Z, b2 - 4ac = △ with 0 ≤ b 〈 √△}of quasi-reduced quadratic forms of fundamental discriminant △. As a matter of fact, we show that for each reduced quadratic form f = aX2 ＋ bXY ＋ cY2 = （a, b, c） of discriminant △〉0（and of sign σ（f） equal to the sign of a）, the quadratic forms associated with f and defined by {〈a＋bu＋cu2,b＋2cu.c〉},with 1≤σ（f）u≤b/2｜c｜ （whenever they exist）, 〈c,-b-2cu,a＋bu＋cu2〉 with b/2｜c｜≤σ（f）u≤[w（f）]=[b＋√△/2｜c｜], are all different from one another and build a set I（f） whose cardinality is #I（f）={1＋[ω（f）],when（2c）｜b,[ω（f）],when （2c）｜b. If f and g are two different reduced quadratic forms, we show that I（f） ∩ I（g） = Ф. Our main result is that the set Q△ is given by the disjoint union of all I（f） with f running through the set of reduced quadratic forms of discriminant △〉0. This allows us to deduce a formula for #（Q△） involving sums of partial quotients of certain continued fractions.