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