| 11. | In particular, an algebraic integer is an integral element of a finite extension K / \ mathbb { Q }.
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| 12. | A special case of-rational points are those that belong to a ring of algebraic integers existing inside the field.
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| 13. | The integral closure is nothing else than the ring \ overline { \ textbf { Z } } of all algebraic integers.
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| 14. | Since the eigenvalues of a matrix are the roots of the polynomial, the eigenvalues of an integer matrix are algebraic integers.
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| 15. | While every number ring is a Dedekind domain, their union, the ring of algebraic integers, is a Pr�fer domain.
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| 16. | In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings.
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| 17. | In particular, in a B�zout domain, prime ( but as the algebraic integer example shows, they need not exist ).
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| 18. | The set of all algebraic integers,, is closed under addition and multiplication and therefore is a commutative subring of the complex numbers.
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| 19. | However there is a unique factorization for ideals, which is expressed by the fact that every ring of algebraic integers is a Dedekind domain.
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| 20. | For instance the ring of analytic functions on any noncompact Riemann surface is a B�zout domain,, and the ring of algebraic integers is B�zout.
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