Primitive And $ Primitive Polynomial Over Finite Fields

This paper mainly deals with distribution of primitive polynomials and primitive elements over finite fields. LetFq be a finite field of q elements with characteristic p , xn + a1xn-1 + ... + an-1x + an be a polynomial over Fq . In bibliography[10] and [11], Han has stated that when n > 7 , there exist primitive polynomials of degree n with a1, a2 prescribed . By making use of two kinds of exponential sums and Cohen's sieve, we better the lower bound concerning the number of primitive elements and also prove the existence of primitive polynomials with a1,a2 prescribed when n = 5,6. In the remaining of the paper we study primitive elements in the form of a + a-1 over finite fields. A counting formula is given as a result, as well as all the finite fields that do not have such elements.