algebraic integer in a sentence

Examples

1. In particular, an algebraic integer is an integral element of a finite extension K / \ mathbb { Q }.
2. A special case of-rational points are those that belong to a ring of algebraic integers existing inside the field.
3. The integral closure is nothing else than the ring \ overline { \ textbf { Z } } of all algebraic integers.
4. Since the eigenvalues of a matrix are the roots of the polynomial, the eigenvalues of an integer matrix are algebraic integers.
5. While every number ring is a Dedekind domain, their union, the ring of algebraic integers, is a Pr�fer domain.
6. In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings.
7. In particular, in a B�zout domain, prime ( but as the algebraic integer example shows, they need not exist ).
8. The set of all algebraic integers,, is closed under addition and multiplication and therefore is a commutative subring of the complex numbers.
9. However there is a unique factorization for ideals, which is expressed by the fact that every ring of algebraic integers is a Dedekind domain.
10. 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.