Sentence:  This paper completes an efficient proof of the Hansen-Mullen Primitivity Conjecture (HMPC) when "n" = 5, 6, 7 or 8. The HMPC (1992) asserts that, with some (mostly obvious) exceptions, there exists a <WSD>primitive polynomial</WSD> of degree "n" over any finite field with any coefficient arbitrarily prescribed. 
Meaning of the word marked with <WSD> tags:  A polynomial over a given finite field whose roots are primitive elements; "especially", the minimal polynomial of a primitive element of said finite field.