A Theorem on Congruences

Let f(x) denote a polynomial with rational integral coefficients and discriminant D. In this note we prove the theorem that if

f(x) = 0 (mod p^r), (1.1)

is solvable for r - δ + 1, where p^δ is the highest power of the prime p dividing D, then the congruence is solvable for all r. While this theorem is contained in a more general theorem of Hensel [1, p. 68], a direct proof seems of interest. Also Theorem 1 below is perhaps of interest in itself.