Solving X^q+1+X+a=0 over Finite Fields
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Solving the equation P_a(X):=X^q+1+X+a=0 over 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.>, 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 identify all the Q-zeros, even in the case p=2, it was solved only under the condition (n, k)=1<cit.>. We discuss this equation without any restriction on p and (n,k). New criteria for the number of the Q-zeros of P_a(x) are proved. For the cases of one or two Q-zeros, we provide explicit expressions for these rational zeros in terms of a. For the case of p^(n, k)+1 rational zeros, we provide a parametrization of such a's and express the p^(n, k)+1 rational zeros by using that parametrization.
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