The Existence Of Special Elements Over Finite Fields

The theory of finite fields is a branch of modern algebra. The origin of the subject reach back intothe17thand18thcentury, and has come to the fore in the last50years because of its diverseapplications in combinatorics, coding theory, and the mathematical study of switching circuits, amongothers. The primitive elements and normal elements, which are special elements over finite fields, areimportant to lucubrate finite fields. Therefore, many researchers are interested in the theory ofprimitive elements and normal elements over finite fields.Let q be the power of a prime number, and n is a positive number. Let GF （qⁿ）denote the finitefield with qⁿ elements. In this thesis, we consider the existence of some specific elements overfinite field GF （qⁿ）.The main results obtained in this paper are listed as follows:（1） There is an element in GF （qⁿ）such that both ξ and ξ+ξ^-1 are all primitive elements ofGF（qⁿ） if q=2^s, and n is an odd number no less than13and s﹥4;（2） For q=2^s, and any odd n, there is an element ξ in GF（qⁿ）such that is a primitivenormal element over GF（qⁿ） and ξ+ξ^-1 is a primitive element over GF（qⁿ） if eithern|（q-1）, and n≥33, or q-1is not divisible by n, and n≥30, s≥6;（3） There is an element in GF（qⁿ）such that both and ξ+ξ^-1 are primitive elements ofGF （qⁿ）and ξ+ξ^-1 is also a normal element of GF （qⁿ）if q=2^s, and either n|（q-1）,and n≥37, or q-1is not divisible by n, and n≥34, s≥6;（4） More over, if q=2^s, and n is an odd number, then there is an element ξ inGF （qⁿ）suchthat is a primitive normal element and ξ+ξ^-1 is a primitive element of GF （qⁿ）if eithern|（q-1）, n≥257, or q-1is not divisible by n, and n≥34, s≥9.（5） Furthermore, when q≥2and n≥29, there are elements in GF（qⁿ） such that ξ+ξ^-1 is aprimitive element of GF（qⁿ）, and Tr （ξ）=a, Tr (ξ^-1)=b for any pair of prescribeda,b∈GF （q）*.