Hourong Qin

SOME DIOPHANTINE EQUATIONS OVER AND WITH APPLICATIONS TO OF A FIELD

ABSTRACT Through determining all the solutions of and in and of and in we have proved that i) is a subgroup of iff , ii) is a subgroup of iff iii) If then is a subgroup of iff n=2 and which partially confirms a conjecture on posed by Browkin in [1].

The Structure of the Tame Kernels of Quadratic Number Fields (III)

Let F be an imaginary quadratic number field and K 2 O F the tame kernel of F. In this article, we determine all possible values of r 4(K 2 O F ) for each type of imaginary quadratic number field F. In particular, for each type of imaginary quadratic number field we give the maximum possible value of r 4(K 2 O F ) and show that each integer between the lower and upper bounds occurs as a value of the 4-rank of K 2 O F for infinitely many imaginary quadratic number fields F.

Computing the Tame Kernel of ℚ(ζ8)

Abstract In this paper, it is proved that the tame kernel of ℚ(ζ8) is trivial.

Higher K-groups of smooth projective curves over finite fields

Abstract Let X be a smooth projective curve over a finite field F with q elements. For m ⩾ 1 , let X m be the curve X over the finite field F m , the m-th extension of F . Let K n ( m ) be the K-group K n ( X m ) of the smooth projective curve X m . In this paper, we study the structure of the groups K n ( m ) . If l is a prime, we establish an analogue of Iwasawa theorem in algebraic number theory for the orders of the l-primary part K n ( l m ) { l } of K n ( l m ) . In particular, when X is an elliptic curve E defined over F , our method determines the structure of K n ( E ) . Our results can be applied to construct an efficient DL system in elliptic cryptography.

Boolean Algebras, Generalized Abelian Rings, and Grothendieck Groups

ABSTRACT A ring R is called generalized Abelian if for each idempotent e in R, eR and (1 − e)R have no isomorphic nonzero summands. The class of generalized Abelian rings properly contains the class of Abelian rings. We denote by GAERS − 1 the class of generalized Abelian exchange rings with stable range 1. In this article we prove, by introducing Boolean algebras, that for any R ∈ GAERS − 1, the Grothendieck group K 0(R) is always an Archimedean lattice-ordered group, and hence is torsion free and unperforated, which generalizes the corresponding results of Abelian exchange rings. Our main technical tool is the use of the ordered structure of K 0(R)+, which provides a new method in the study of Grothendieck groups.

ON K 2 OF DIVISION ALGEBRAS

ABSTRACT In this paper, it is proved that if F is a global field, then for any integer n > 3, there is an extension field E over F of degree n such that K 2 E is not generated by the Steinberg symbols {a, b} with a ∈ F*, b, ∈ E*. If however, F is a number field and D is a finite-dimensional central division F-algebra with square free index, then K 2 D is always generated by the Steinberg symbols {a, b} with a ∈ F*, b ∈ D*. Finally, the tame kernels of central division algebras over F are expressed explicitly.

Imaginary Quadratic Fields with Ono Number 3

In this article, we prove that an imaginary quadratic field F has the ideal class group isomorphic to ℤ/2ℤ ⊕ ℤ/2ℤ if and only if the Ono number of F is 3 and F has exactly 3 ramified primes under the Extended Riemann Hypothesis (ERH). In addition, we give the list of all imaginary quadratic fields with Ono number 3.

Epimorphisms in the Category of ℓ-Groups

In the category of l-groups, we introduce the concepts of Hopfian and generalized Hopfian l-groups. An l-group G is called Hopfian if every surjective l-homomorphism f : G → G is an l-isomorphism, and G is called generalized Hopfian if every surjective l-homomorphism f : G → G has a small kernel in G. By an example we show that the class of generalized Hopfian l-groups is a proper subclass of Hopfian l-groups. In this paper, we establish some characterizations for them, which generalize some results in [9].

Regularn-simplices in ℝn with vertices in ℤn

In this paper theI andII regularn-simplices are introduced. We prove that the sufficient and necessary conditions for existence of anI regularn-simplex in ℝn are that ifn is even thenn = 4m(m + 1), and ifn is odd thenn = 4m + 1 with thatn + 1 can be expressed as a sum of two integral squares orn = 4m - 1, and that the sufficient and necessary condition for existence of aII regularn-simplex in ℝn isn = 2m2 - 1 orn = 4m(m + 1)(m ∈ ℕ). The connection between regularn-simplex in ℝn and combinational design is given.