I understand that an integral domain $D$ is said to be integrally closed in $S$ if whenever an element $s \in S$ can be viewed as a root of a polynomial with coefficients in $D$, it must be in $D$.
However, apparently $\mathbb Z$ is integrally closed in $\mathbb Q$, which I don't get, since surely given any $\frac{a}{b} (a, b \in \mathbb Z$), it is the root of the polynomial $bx – a$, and yet $\frac{a}{b}$ may not be in $\mathbb Z$.
Where have I gone wrong here?
Best Answer
The polynomial $\frac{a}{b}$ satisfies is required to be monic, namely the coefficient of the highest degree term has to be $1$.