In the book Principle in Algebraic Geometry of Griffiths & Harris, I encountered the following definition of the resultant of two polynomials:
If $R$ is a UFD and $u,v\in R[t]$ are relatively prime, then there exist relatively prime elements $\alpha,\beta\in R[t]$, $\gamma\neq 0\in R$, such that $$\alpha u+\beta v=\gamma.$$ $\gamma$ is called the resultant of $u$ and $v$.
This definition seems strange to me, since I'm much more familiar with the definition of resultant using determinants of the coefficients of two polynomials. However, I tried to show how these definitions are equivalent with each other. It is clear that $\gamma\in (u,v)$, the ideal of $R[t]$ generated by $u$ and $v$. Then $(u,v)\cap R$ is non-empty and different from $(0)$. Why this must be true? And why $\gamma$ must be unique?
Hope someone can explain that to me. Thanks in advance.