1. The following are equivalent definitions of an algebraic integer . 2. Algebraic integers are a special case of integral elements of a ring extension.3. Namely, the same power of q divides each of these algebraic integers . 4. Consider the ring of algebraic integers in Q ( sqrt ( 2 ) ). 5. Here and are ordinary integers, whereas the factors are algebraic integers in the cyclotomic field. 6. In fact, the ( rational ) integers are the algebraic integers that are also rational numbers. 7. This polynomial transformation is often used to reduce questions on algebraic numbers to questions on algebraic integers . 8. By multiplying A by proper integer if necessary we may assume that A is an algebraic integer . 9. Again, since the quadratic Gauss sum is an algebraic integer , we can use modular arithmetic with it. 10. Roots of unity are algebraic integers , hence their J . 15 : 13, 27 April 2009 ( UTC)
