Trinomial Equations of Degree 6 over

Finding roots of a single variable polynomial is among the oldest problems of mathematics. This problem is solved in the field of reals but was paid less attention in the field of -adic numbers, the counterpart of the field of reals. Recently, this problem has been raised up again in considering the -adic lattice models of statistical mechanics. We introduce a cube root function over the -adic field , which enables us to explicitly prescribe the roots of the trinomial equation of degree 6 over . Namely, we calculate the -adic absolute value and the first digit of roots of the trinomial equation of degree 6 over .