symmetric monoidal (∞,1)-category of spectra
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 $K$.