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Solving equations over finite fields is an important problem from both theoretical and practice points of view. The problem of solving explicitly the equation P a ( X ) = 0 over the finite field F Q , where P a ( X ) : = X q + 1 + X + a , Q = p n , q = p k , a ∈ F Q ⁎ and p is a prime, arises in many different contexts including finite geometry, the inverse Galois problem [1] , the construction of difference sets with Singer parameters [9] , determining cross-correlation between m-sequences [10] and to construct error correcting codes [5] , cryptographic APN functions [6] , [7] , designs [21] , as well as to speed up the index calculus method for computing discrete logarithms on finite fields [11] , [12] and on algebraic curves [18] . In fact, the research on this specific problem has a long history of more than a half-century from the year 1967 when Berlekamp, Rumsey and Solomon [2] firstly considered a very particular case with k = 1 and p = 2 . In this article, we discuss the equation P a ( X ) = 0 without any restriction on p and gcd ( n , k ) . In a very recent paper [15] , the authors have left open a problem that could definitely solve this equation. More specifically, for the cases of one or two F Q -zeros, explicit expressions for these rational zeros in terms of a were provided, but for the case of p gcd ( n , k ) + 1 F Q − zeros it was remained open to compute explicitly the zeros. This paper solves the remained problem, thus now the equation X p k + 1 + X + a = 0 over F p n is completely solved for any prime p, any integers n and k. |
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