Is the integral closure of local domain a local ring?

Suppose $A$ is a local domain, with field of fractions $K$, let $A’$ be the integral closure of $A$ in $K$, is $A’$ a local ring?

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Not necessarily. E.g. consider $A = \mathbb Z_{(5)} + 5 \mathbb Z_{(5)}[i]$. (Here $\mathbb Z_{(5)}$ denotes $\mathbb Z$ localized at the prime ideal $(5)$.)
Then $A’ = \mathbb Z_{(5)}[i],$ which has two maximal ideals (the ideals generated by $2 \pm i$).


For a geometric example, consider the local ring at the node of the nodal cubic
$y^2 = x^2(x +1).$