Let $R$ be the ring of polynomials over a field $\mathbb{F}$. Let $p_1,p_2,\dots, p_n$ be elements of $R$. Prove that the greatest common divisor of $p_1,p_2,\dots, p_n$ is $1$ if and only if there is an $n\times n$ matrix over $R$ of determinant $1$ whose first row is $(p_1,p_2,\dots, p_n)$.
A set of polynomials can have GCD $1$ if they have no common roots among themselves in $\mathbb{F}$. But I don't understand how that relates with a matrix of determinant $1$ with first row consisting those polynomials. Any help is appreciated.