Abstract Algebra: Theory and Applications · Problem 17.5.18

Integral Domains — Thomas Judson

Mathematics — Abstract Algebra · rational-root-theorem

Problem

The Rational Root Theorem. Let p(x)=anxn+an−1xn−1+⋯+a0∈Z[x],p(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_0 \in \mathbb{Z}[x], where an≠0a_n \neq 0. Prove that if p(r/s)=0p(r/s) = 0, where gcd⁡(r,s)=1\gcd(r,s) = 1, then r∣a0r \mid a_0 and s∣ans \mid a_n.

Related theory

Browse the problem library