Doubt about a paragraph in the book "Algebraic Number Theory by Neukirch".

Question

Please refer to the question in the given link.

The question already has an answer here

An element is integral iff its minimal polynomial has integral coefficients.

My question is ---

Here, $A$ is integrally closed in $K$. And $\beta_i$'s $\in \bar{K}$. Don't we have to show that $\beta_i$'s $\in K$ to show they belong to $A$? Because $A$ is integrally closed in $K$ not in $\bar{K}$.

Thank you.

Answer 1

Here, $\beta_i$'s belong to $\bar{K}$ which are integral over $A$. So, coefficients of $p(x)$ are integral over $A$. But the coefficients of $p(x)$ belong to $K$. So, coefficients of $p(x)$ belong to $A$ as $A$ is integrally closed in $K$. Hence, $p(x)\in A[x]$.