Can we replace the $B$ to $A$ in this proposition

Question

I am working through Atiyah's Commutative algebra and am having question with the following proposition:

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Proposition 5.15. Let $A$ $\subseteq$ $B$ be integral domains, $A$ integrally closed, and let $x$ $\in$ $B$ be integral over an ideal $a$ of $A$. Then $x$ is algebraic over the field of fractions $K$ of $A$, and if its minimal polynomial over $K$ is $t^n+a_1t^{n-1}+\ldots+a_n$, then $a_1,\ldots, a_n$ lie in $r(a)$.

1.Can we replace the $x$ $\in$ $B$ to $x$ $\in$ $A$ in this proposition? Because $A$ integrally closed.

2.Why we need the polynomial be the minimal polynomial?

Thanks in advance.

Answer 1

1, $A$ is integrally closed means $A$ is closed in the field of fractions, not in $B$.

2, Because conjugate elements are other roots of the minimal polynomial of $x$. See: here