Definition:Algebraic Integer
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Definition
Let $K/\Q$ be a number field.
We call $\alpha \in K$ an algebraic integer if it satisfies a monic polynomial $f \in \Z[X]$.
The set of all algebraic integers is usually denoted $\mathfrak o_K$ or $\mathcal O_K$.
By Ring of Algebraic Integers it is a ring, hence usually referred to as the ring of algebraic integers of $K$.
Quadratic Integers
If $K/\Q$ has degree two, then an algebraic integer in $K$ is called a quadratic integer.
Examples incluse the Gaussian integers and Eisenstein integers.
