Solution of Equations of the Third Degree in a Field of Characteristic 3

Polynomials of the third degree on $GF(3^k)$ are considered. A condition is deduced for which $GF(3^k)$ will be the field of the expansion of a given polynomial. It is shown that to find the roots of such polynomials it is sufficient to solve a linear system with $k-1$ unknowns. In the cases $k=3,4,5$ explicit expressions are also given for the roots in terms of the coefficients. The results explained can be used for decoding triple Bose–Chaudhuri codes correcting three errors [W. W. Peterson, Error-Correcting Codes, Cambridge, M.I.T. Press, 1961; M. V. Matveeva, Probl. Peredachi Inf., 1968, vol. 4, no. 1, pp. 20–27].