Let F_q be a finite field of characteristic p. In this paper, by using the index sum method the authors obtain a sufficient condition for the existence of a primitive elementα∈ F_(q^n) such that α + α^(-1)is also ...Let F_q be a finite field of characteristic p. In this paper, by using the index sum method the authors obtain a sufficient condition for the existence of a primitive elementα∈ F_(q^n) such that α + α^(-1)is also primitive or α + α^(-1)is primitive and α is a normal element of F_(q^n) over F_q.展开更多
Let q be a prime or prime power and Fq^n the extension of q elements finite field Fq with degree n （n 〉 1）. Davenport, Lenstra and School proved that there exists a primitive element α ∈ Fq^n such that α generat...Let q be a prime or prime power and Fq^n the extension of q elements finite field Fq with degree n （n 〉 1）. Davenport, Lenstra and School proved that there exists a primitive element α ∈ Fq^n such that α generates a normal basis of Fq^n over Fq. Later, Mullin, Gao and Lenstra, etc., raised the definition of optimal normal bases and constructed such bases. In this paper, we determine all primitive type I optimal normal bases and all finite fields in which there exists a pair of reciprocal elements α and α^-1 such that both of them generate optimal normal bases of Fq^n over Fq. Furthermore, we obtain a sufficient condition for the existence of primitive type II optimal normal bases over finite fields and prove that all primitive optimal normal elements are conjugate to each other.展开更多
For a prime p and a positive integer k,let q=p^(k) and F_(q)^(n) be the extension field of F_(q).We derive a sufficient condition for the existence of a primitive element α in F_(q)^(n) such that α^(3)-α+1 is also ...For a prime p and a positive integer k,let q=p^(k) and F_(q)^(n) be the extension field of F_(q).We derive a sufficient condition for the existence of a primitive element α in F_(q)^(n) such that α^(3)-α+1 is also a primitive element of F_(q)^(n) ,a sufficient condition for the existence of a primitive normal element a in F_(q)^(n) over F_(q) such that α(3)-α+1 is a primitive element of F_(q)^(n) ,and a suficient condition for the existence of a primitive normal element a in F_(q)^(n) over F_(q) such that а^(3)-а+1 is also a primitive normal element of F_(q)^(n) over F_(q).展开更多
In [5, 6] the relationships between test words, generic elements, almost primitivity and tame almost primitivity were examined in free groups. In this paper we extend the concepts and connections to general free produ...In [5, 6] the relationships between test words, generic elements, almost primitivity and tame almost primitivity were examined in free groups. In this paper we extend the concepts and connections to general free products and in particular to free products of cyclic groups.展开更多
Let H be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed feld of characteristic zero.We describe all annihilator ideals of indecomposable H-modules by generators.In particular,we giv...Let H be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed feld of characteristic zero.We describe all annihilator ideals of indecomposable H-modules by generators.In particular,we give the classification of all ideals of the finite-dimensional pointed rank one Hopf algebra of nilpotent type over the Klein 4-group.展开更多
基金supported by the National Natural Science Foundation of China(No.11401408)the Natural Science Foundation of Sichuan Province(No.14ZA0034)+2 种基金the Sichuan Normal University Key Project Foundation(No.13ZDL06)supported by the National Natural Science Foundation of China(No.11001170)the Natural Science Foundation of Shanghai Municipal(No.13ZR1422500)
文摘Let F_q be a finite field of characteristic p. In this paper, by using the index sum method the authors obtain a sufficient condition for the existence of a primitive elementα∈ F_(q^n) such that α + α^(-1)is also primitive or α + α^(-1)is primitive and α is a normal element of F_(q^n) over F_q.
基金Supported by the National Natural Science Foundation of China (Grant No10990011)Special Research Found for the Doctoral Program Issues New Teachers of Higher Education (Grant No20095134120001)the Found of Sichuan Province (Grant No09ZA087)
文摘Let q be a prime or prime power and Fq^n the extension of q elements finite field Fq with degree n （n 〉 1）. Davenport, Lenstra and School proved that there exists a primitive element α ∈ Fq^n such that α generates a normal basis of Fq^n over Fq. Later, Mullin, Gao and Lenstra, etc., raised the definition of optimal normal bases and constructed such bases. In this paper, we determine all primitive type I optimal normal bases and all finite fields in which there exists a pair of reciprocal elements α and α^-1 such that both of them generate optimal normal bases of Fq^n over Fq. Furthermore, we obtain a sufficient condition for the existence of primitive type II optimal normal bases over finite fields and prove that all primitive optimal normal elements are conjugate to each other.
基金This work was funded by the Council of Scientific and Industrial Research,New Delhi,Government of India’s research grant no.09/796(0099)/2019-EMR-I.
文摘For a prime p and a positive integer k,let q=p^(k) and F_(q)^(n) be the extension field of F_(q).We derive a sufficient condition for the existence of a primitive element α in F_(q)^(n) such that α^(3)-α+1 is also a primitive element of F_(q)^(n) ,a sufficient condition for the existence of a primitive normal element a in F_(q)^(n) over F_(q) such that α(3)-α+1 is a primitive element of F_(q)^(n) ,and a suficient condition for the existence of a primitive normal element a in F_(q)^(n) over F_(q) such that а^(3)-а+1 is also a primitive normal element of F_(q)^(n) over F_(q).
文摘In [5, 6] the relationships between test words, generic elements, almost primitivity and tame almost primitivity were examined in free groups. In this paper we extend the concepts and connections to general free products and in particular to free products of cyclic groups.
基金supported by the National Natural Science Foundation of China(Grant No.12371041).
文摘Let H be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed feld of characteristic zero.We describe all annihilator ideals of indecomposable H-modules by generators.In particular,we give the classification of all ideals of the finite-dimensional pointed rank one Hopf algebra of nilpotent type over the Klein 4-group.
文摘Golomb猜想(参看Golomb,S.,Algebraic Construction for Cost as Arrays,to be Published in Journal of Combinatorial Theory)为:在任何有限域GF(P^(2))中总存在两个本原元,它们的和等于1。
