Topology of Numbers · Problem 6.2.2

Representations by Quadratic Forms — Allen Hatcher

Mathematics — Number Theory · polynomial-congruences-chinese-remainder

Problem

The following is a generalization of Lemma 6.4. Let

P(x)

be a polynomial with integer coefficients and let

n

be an integer. Show that if the congruence

P(x) \equiv n

has a solution mod

m_1

and also a solution mod

m_2

where

m_1

and

m_2

are coprime, then it has a solution mod

m_1 m_2

. Give an example where this fails without the coprimeness condition.