A Theorem on Congruences


Affiliations

  • Duke University, Durham, United States

Abstract

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 pr), (1.1)

is solvable for r - δ + 1, where pδ 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.


Subject Discipline

Mathematical Sciences

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