If \(f\) and \(g\) are independent polynomials over a field \(F\) and \(h\) is a non-zero polynomial over \(F\), show that \(fh\) and \(gh\) are independent.
Solution
Suppose \[ a(fh)+b(gh)=0 \] for scalars \(a,b\in F\). Then \[ (af+bg)h=0. \] Since \(F[x]\) is an integral domain and \(h\ne0\), it follows that \[ af+bg=0. \] Because \(f\) and \(g\) are linearly independent, \(a=b=0\). Hence \(fh\) and \(gh\) are linearly independent.