Exercise 29, § 1

Let A be an integrally closed domain and K its field of fractions. (a) Let L be an algebraic extension of K and B the integral closure of A in L. Show that for every fractional ideal a of A, if we write $b = aB, b \cap A = a''$. (Reduce it to the case where $A \subset \tilde{a}$, in other words A: $a \subset A$ and prove that B: $b \subset B$; for this, let $x \in B$: $b$ and let $c_i (1 \leq i \leq n)$ be the coefficients of its minimal polynomial over K; note that for ally $\in a$ the elements $c_i y^i (1 \leq i \leq n)$ belong to A and deduce that the $c_i$ belong to A.) (b) Let C be a polynomial ring $A[X_\lambda]_{\lambda \in L}$ in any family of indeterminates. Show that for every fractional ideal a of A, if we write $c = aC, c \cap A = a''$ (same method).