Let \(F\) be a field. Show that the ideal generated by a finite number of polynomials \(f_1, \ldots, f_n\) in \(F[x]\) is the intersection of all ideals containing \(f_1, \ldots, f_n\).
Solution
Let \[ J=(f_1,\ldots,f_n) =\left\{\sum_{i=1}^n h_i f_i:h_i\in F[x]\right\}. \] Then \(J\) is an ideal containing every \(f_i\).
If \(I\) is any ideal containing \(f_1,\ldots,f_n\), then for arbitrary \(h_i\in F[x]\) each product \(h_if_i\) lies in \(I\), and therefore so does their sum. Hence \[ J\subseteq I. \] Thus \(J\) is contained in every ideal containing all the \(f_i\). Since \(J\) itself is one such ideal, \[ J=\bigcap_{\substack{I\text{ ideal}\\ f_1,\ldots,f_n\in I}} I. \]