# 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?

#### Solutions Collecting From Web of "Is the integral closure of local domain a local ring?"

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
$y^2 = x^2(x +1).$