Recovering an algebraic number from its residue modulo an ideal is not difficult if this ideal is generated by a rational integer. Often, however, one needs to work with a prime ideal not of this form. When the algebraic number is an integral linear combination of powers of a root &agr; of a monic irreducible polynomial over &ZZ;, this problem has been treated by Lenstra using an algorithm for finding short vectors in lattices [1]. This paper extends Lenstra’s method by treating the general case of a rational linear combination of powers of &agr;. The author also extends this method to the situation where the number field is generated by several generators &agr;. Finally, he explains how the method can work if a reducible polynomial is given for an algebraic generator.