Corollary 6

— Let $f, g_1, g_2, ..., g_n \in K[X]$. If f is prime to $g_1, g_2, ..., g_n$, then f is prime to $g_1g_2...g_n$.

\* COROLLARY 7. — For f and g to be relatively prime it is necessary and sufficient that they have no common roots in any extension of K. For if d is a gcd of f, g then the roots common to f and g in an extension K' of K are the roots of d in K'. Now the corollary follows from V, p. 21, Prop. 4.