\begin{definition}[Definition:Algebraic Integer]
Let $K / \Q$ be an algebraic number field.
{{explain|The notation at this point needs to be expanded and explained. The concept is pitched at a level more abstract than can easily be comprehended by someone who is merely exploring for example $\Z \sqbrk {\sqrt 2}$. It is also worth adding further definitions which may serve to bring it down to earth.}}
Then $\alpha \in K$ is an '''algebraic integer''' {{iff}} it satisfies a monic polynomial $f \in \Z \sqbrk X$.
The set of all '''algebraic integers''' in $K$ is denoted $\mathfrak o_K$ or $\OO_K$.
By Ring of Algebraic Integers it is a ring, hence usually referred to as the '''ring of algebraic integers''' of $K$.
\end{definition}