# Complete solution over p^n of the equation X^p^k+1+X+a=0

The problem of solving explicitly the equation P_a(X):=X^q+1+X+a=0 over the finite field Q, where Q=p^n, q=p^k and p is a prime, arises in many different contexts including finite geometry, the inverse Galois problem <cit.>, the construction of difference sets with Singer parameters <cit.>, determining cross-correlation between m-sequences <cit.> and to construct error correcting codes <cit.>, cryptographic APN functions <cit.>, designs <cit.>, as well as to speed up the index calculus method for computing discrete logarithms on finite fields <cit.> and on algebraic curves <cit.>. Subsequently, in <cit.>, the Q-zeros of P_a(X) have been studied. In <cit.>, it was shown that the possible values of the number of the zeros that P_a(X) has in Q is 0, 1, 2 or p^(n, k)+1. Some criteria for the number of the Q-zeros of P_a(x) were found in <cit.>. However, while the ultimate goal is to explicit all the Q-zeros, even in the case p=2, it was solved only under the condition (n, k)=1 <cit.>. In this article, we discuss this equation without any restriction on p and (n,k). In <cit.>, for the cases of one or two Q-zeros, explicit expressions for these rational zeros in terms of a were provided, but for the case of p^(n, k)+1 Q- zeros it was remained open to explicitly compute the zeros. This paper solves the remained problem, thus now the equation X^p^k+1+X+a=0 over p^n is completely solved for any prime p, any integers n and k.

READ FULL TEXT