The translate and line properties for 2-primitive elements in quadratic extensions.

Let r , n > 1 be integers and q be any prime power q such that r   |   q n − 1. We say that the extension 𝔽 q n / 𝔽 q possesses the line property for r -primitive elements if, for every α , 𝜃 ∈ 𝔽 q n ∗ , such that 𝔽 q n = 𝔽 q (𝜃) , there exists some x ∈ 𝔽 q , such that α (𝜃 + x) has multiplicative order (q n − 1) / r. Likewise, if, in the above definition, α is restricted to the value 1 , we say that 𝔽 q n / 𝔽 q possesses the translate property. In this paper, we take r = n = 2 (so that necessarily q is odd) and prove that 𝔽 q 2 / 𝔽 q possesses the translate property for 2-primitive elements unless q ∈ { 5 , 7 , 1 1 , 1 3 , 3 1 , 4 1 }. With some additional theoretical and computational effort, we show also that 𝔽 q 2 / 𝔽 q possesses the line property for 2-primitive elements unless q ∈ { 3 , 5 , 7 , 9 , 1 1 , 1 3 , 3 1 , 4 1 }. [ABSTRACT FROM AUTHOR]