Suppose there is a (algebraic) number field $K$.
Algebraic integer is a root of some monic polynomial with coefficients in $\mathbb{Z}$.
The elements of $K$ are the root of some monic polynomial with coefficients in $\mathbb{Q}$.
As the field is commutative, every monic equation with coefficients in $\mathbb{Q}$ can be converted into monic equation with coefficients in $\mathbb{Z}$ (By multiplying least common multiples of denominators in coefficients.) and have the same roots.
This suggests that every number in $K$ is an algebraic integer.
Is this right understanding? If not, where did I go wrong?