题目：On the existence of completely normal elements with some special properties over finite fields

报告人：曹喜望 教授，南京航空航天大学

时间：2021年10月22日（星期五）14:30---15:30

腾讯会议 ID：561 395 470

会议密码：202110

摘要：An element of F_q^n is said to be completely normal over F_q if it is simultaneously normal over F_q^l for all l dividing n. It is known that for any q and n, there exist the completely normal elements of F_q^n. Recently,Huczynska, Mullen, Panario and Thomson (2013) introduced the concept of k-normal elements, as a generalization of normal elements. For 0≤ k ≤ n, the element \xi of F_q^n is said to be a k-normal element if all the conjugates of  \xi span a vector space of dimension n-k over F_q. In this talk, we first give a sufficient condition for the existence of a completely normal element  of F_q^n over Fq such that \xi^q-\xi is primitive 1-normal. We also provide some bounds for the number of completely normal elements of F_q^n over F_q. Subsequently, using the obtained results we prove that if n is odd and q -1\geq  n\geq  7, or n is even and q -1 \geq n\geq  8, then there exists a completely normal element  of F_q^n over F_q such that \xi^q-\xi is primitive 1-normal. This is a joint work with Hanglong Zhang.