### SEMINAR: Existence theorems for r-primitive elements in finite fields19-10-2020

**Speaker: ** Prof. Stephen D. Cohen,Emeritus Professor at the University of Glasgow

**Title: **Existence theorems for r-primitive elements in finite fields

**Date/Time:** 21 October 2020 13:40 - 2:30 pm

**Zoom: Meeting ID**: 916 5812 4356 **Passcode**: Algebra

**Abstract:**Let r|q-1. An element of F_q is r-primitive if it has order (q -1)/r. Thus, a primitive element is 1- primitive and an r-primitive element is the rth power of a primitive element of F_q. We describe some existence theorems for general r-primitive elements and, in particular, analogues for 2-primitive elements of the following complete existence theorems for primitive elements.

Theorem A (1990). For any n ≥ 2 and a ∈ F_q (necessarily with a ≠ 0 if n = 2) there exists a primitive α ∈ F_(q^n ) with trace a over F_q, except when a = 0, n = 3, q = 4.

Theorem B (1983). Every line in F_(q^2 ) contains a primitive element. (A line in F_(q^2) is a set of the form {β(γ + a)∶ a ∈ F_q}, for some nonzero β ∈ F_(q^2 ) ,γ ∈ F_(q^2 ) \ F_q.). Joint work with Giorgos Kapetanakis.

**Bio: ** Stephen D. Cohen is a mathematician, Emeritus Professor at the University of Glasgow, Scotland. Professor Cohen obtained his Ph. D. in 1969 from the University of Glasgow where he became a faculty member in 1968, progressing eventually to becoming Professor of Number Theory in 2002. During that period he had sabbaticals at the University of Illinois at Urbana (1978), the University of the Witwatersrand, Johannesburg (1986), the University of Tasmania (1993), and Bowling Green State University, Ohio (1994). Professor Cohen was the main organiser of the Third International Conference on Finite Fields and Applications at the University of Glasgow (1995) and has been on the Editorial Board of several journals. He is the author of around 130 published papers from 1968 onwards, mostly on finite fields but also on subjects like the distribution of Galois groups, free groups from fields, semifields, and classical number theory.