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Given a unital ring $k$, a monic polynomial over $k$ is a polynomial with coefficients in $k$, whose highest order coefficient is $1$.
A root of a monic polynomial over $k$ is by definition an algebraic integer over $k$.
Here algebraic integer usually means algebraic integer over $\mathbf{Z}$. All algebraic integers form a field called the integral closure of $\mathbf{Z}$ in $\mathbf{C}$.
On the other hand, for a number field $K$, an integer in $K$ is an algebraic integer over $\mathbf{Z}$ which is in $K$; all integers in $K$ form a ring of integers $\mathcal{O}_K$ of the number field $K$.
Last revised on October 2, 2012 at 12:58:56. See the history of this page for a list of all contributions to it.