Solution to a Problem of Lubelski and an Improvement of a Theorem of His

摘要

Summary. The paper consists of two parts, both related to problems of Lubelski, but unrelated otherwise. Theorem 1 enumerates for a = 1,2 the finitely many positive integers D such that every odd positive integer L that divides x~2 +Dy~2 for (x, y) = 1 has the property that either L or 2~aL is properly represented by x~2 + Dy~2. Theorem 2 asserts the following property of finite extensions k of Q: if a polynomial f ∈ k[x] for almost all prime ideals p of k has modulo p at least v linear factors, counting multiplicities, then either f is divisible by a product of v + 1 factors from k[x] k, or f is a product of v linear factors from k[x].