151 Euclidean algorithm

Proof. If $P$ and $Q$ have a common zero $\lambda$, then

contrary to the equality. Conversely, suppose that $P$ and $Q$ have no common zeros and carry out the Euclidean algorithm for $P$ and $Q$ to obtain complex polynomials $M,N,R$ such that $R$ is the highest common factor of $P$ and $Q$ and $PM+QN=R$. If $R=r$ is a nonzero constant, then we can multiply through by $r^{-1}$ to obtain $PM/r+QN/r=1$, as required. Otherwise, $R$ is a complex polynomial of positive degree, and hence by the fundamental theorem of algebra has a complex zero $\lambda$. Now $R$ is a factor of both $P$ and of $Q$, so $s-\lambda$ is a factor of both $P$ and $Q$, so $\lambda$ is a common zero of $P$ and $Q$, contrary to assumption.