Let $k$ be a field. Let $a_1,\cdots,a_n$ be algebraic over $k$. Then $k[a_1,\cdots,a_n]=k(a_1,\cdots,a_n)$. Let $f_i(a_1,\cdots,a_{i-1},X_i)$ be the minimal polynomial of $a_i$ over $k(a_1,\cdots,a_{i-1})$. Suppose that $R_{i}(X_1,\cdots,X_n) \in k[X_1,\cdots,X_n]$ such that $R_i(a_1,\cdots,a_i,X_{i+1},\cdots,X_n)=0$. Then why is it true that $f_i(a_1,\cdots,a_{i-1},X_i)$ divides $R_i(a_1,\cdots,a_{i-1},X_i,\cdots,X_n)$? What confuses me is that $R_i(a_1,\cdots,a_{i-1},X_i,\cdots,X_n)$ is a polynomial in many variables $X_i,\cdots,X_n$.
Motivation: Matsumura, Commutative Ring Theory, proof of Theorem 5.1 page 32.