Could someone help me on the following question?
Let $F$ be an extension of a field $K$. Let $P_{1}(X),P_{2}(X)\in K[X]$ and $\alpha\in F$. Show that $\alpha$ it is a common root of $P_{1}(X)$ and $P_{2}(X)$ if and only if $\alpha$ is a root of greatest common divisor of $P_{1}(X)$ and $P_{2}(X)$.
($\Rightarrow$) Just note that greatest common divisor of $P_{1}(X),P_{2}(X)$ is $P_{1}(X)Q_{1}(X)+P_{2}(X)Q_{2}(X)$.
($\Leftarrow$) Here I am in doubt. If $\alpha$ is the root of the greatest common divisor, then $P_{1}(\alpha)Q_{1}(\alpha)+P_{2}(\alpha)Q_{2}(\alpha)=0$. From here I do not know how to conclude that alpha is not the root of $P_{1}(X)$ and $P_{2}(X)$.
($\Rightarrow$) Just note that greatest common divisor of $P_{1}(X),P_{2}(X)$ can be written as $P_{1}(X)Q_{1}(X)+P_{2}(X)Q_{2}(X)$.
($\Leftarrow$) If $\alpha$ is the root of the greatest common divisor, then, since it divides $P_1$ and $P_2$, we get that $\alpha$ is also a root of $P_1$ and $P_2$. (More precisely, if a polynomial $Q$ divides a polynomial $P$, then we can write $P=QR$, and $Q(\alpha)=0$ implies $P(\alpha)=0$.)