Gustav Kjærbye Bagger (UNSW Canberra) - Chasing primitive elements

A primitive element of a Galois field F is a generator of the group of units F*. Finding such elements is entirely non-trivial and this search has given rise to many interesting results in number theory. If one is willing to assume the generalised Riemann hypothesis, then every (non-silly) integer is primitive to infinitely many prime order fields. Moreover, the least primitive element in such fields is less than the square root of the order. I will present an effective method for finding primitive elements in Galois fields with prime-power order. This method is an amalgamation of my hybrid bound on prime divisors and joint work with J. Punch on prime sieves. I will also discuss some of my recent work on least primitive elements.