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If $B$ is an integrally closed domain and $B to A$ is an integral extension, then $A$ is the integral closure of $B$ in $K(A)$ or not?

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Here $K(A)$ and $K(B)$ mean the fraction fields of $A$ and $B$, respectively. I see that the integral closure of $B$ in $K(A)$, denoting it as $C$, satisfies $A subseteq C$ for $Bto A$ being an integral extension. Is it true that $C = A$?

I want to show that if $p/q in K(A)$ satisfies some monic polynomial $x^n + b_{n-1}x^{n-1} + cdots + b_1 x + b_0 = 0$ with $b_i in B$ and $p, q in A$, $q$ not a unit, then $p in (q)$. But I stucked at this point. I think if it's true, $B$ being integrally closed has to be used. Otherwise taking $A = B = k[t^2, t^3]$ is an counterexample. Or is it generally only $A subset C$?

Some kind of an reverse of the current question may be related: fraction field of the integral closure

Thanks a lot for your helps!



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