How can I calculate algebraic integer ( primitive roots of unitary for example) mod p ?
I need them in modular representation of finite groups.
More explanation :
In page 16
I want to find the image of primitive roots of unitary under the natural map.
Given an algebraic number field $K$ there is a concept of algebraic integers of that field usually denoted $\mathcal {O}_K$: this consists of those elements of $K$ satisfying monic equations with integer coefficients.
There is no unique factorization available for the elements of these rings: instead we have a unique factorization of the ideals as product of non-zero prime ideals (they are also maximal ideals).
Now coming to your question given a prime number $p$ the principal ideal generated by it in $\mathcal {O}_K$ need not be a prime ideal; so we factorize this ideal in $\mathcal {O}_K$: the prime ideals occurring in the factorization are precisely those containing the element $p$.
One can talk of quotient rings $\mathcal {O}_K$ modulo such prime ideals $P$. This is the analogue of mod $p$ operation you are looking for. The restriction of the canonical map $\mathcal {O}_K\to \mathcal {O}_K/P$ to $\mathbf{Z}$ will provide an embedding of the finite field of $p$ elements.
$\mathcal {O}_K/P$ is also a finite field and its degree over its prime subfield is called the inertial degree of the prime ideal $P$ over $p$. As there are many prime ideals this depends on the choice of $P$. However for Galois extension $K$ over over rationals these inertial degrees are the same. You can read these things in the book Number Fields by Marcus, for example.