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**TheArtofSymmetry** A corollary from the above definition is that the algebraic integers in $\displaystyle \mathbb{Q}$ are integers $\displaystyle \mathbb{Z}$. For instance, if $\displaystyle \beta$ is an algebraic integer in $\displaystyle \mathbb{Q}$, then the minimal polynomial of $\displaystyle \beta=a/b \in \mathbb{Q}$, where a and b are integers and $\displaystyle b \neq 0$, is $\displaystyle bx - a $. Since $\displaystyle \beta$ is an algebraic integer by hypothesis, $\displaystyle \beta$ should be the root of a monic polynomial with coefficients in $\displaystyle \mathbb{Z}$. Thus b is 1 and $\displaystyle \beta \in \mathbb{Z}$.