A new approach of some problems concerning polynomials over finite fields

In this paper we have obtained some results regarding the number of roots and the irreducibility property of the polynomials which have the form axn + x + 1 over finite fields whose characteristic is 2.Some of these results appear as generalizations of previous results obtained by Berlekamp, Rumsey, and Solomon (1967, On the solution of algebraic equations over finite fields, Information and Control 10, 553‐564).We have also proposed a new method of studying equations in finite fields using the properties of the coefficients of certain characteristic polynomials related to these equations