Let \(p\) be a monic polynomial over the field \(F\), and let \(f\) and \(g\) be relatively prime polynomials over \(F\). Prove that the g.c.d. of \(pf\) and \(pg\) is \(p\).
Solution
Since \(f\) and \(g\) are relatively prime, there exist \(a,b\in F[x]\) such that \[ af+bg=1. \] Multiplying by \(p\) gives \[ a(pf)+b(pg)=p. \] Thus every common divisor of \(pf\) and \(pg\) divides \(p\). On the other hand, \(p\) itself divides both \(pf\) and \(pg\). Hence the monic greatest common divisor of \(pf\) and \(pg\) is exactly \[ \boxed{p}. \]