| 31. | Integral domains are characterized by the condition that they are irreducible ( that is there is only one minimal prime ideal ).
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| 32. | In scheme theory, the spectrum of an integral domain has a unique generic point, which is the minimal prime ideal.
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| 33. | In algebraic geometry, taking the spectrum of a ring whose reduced ring is an integral domain is an irreducible topological space.
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| 34. | They both noticed it was precisely the extra piece of structure needed to turn an integral domain into a principal ideal domain.
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| 35. | A "'Euclidean domain "'is an integral domain which can be endowed with at least one Euclidean function.
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| 36. | An integral domain where a gcd exists for any two elements is called a GCD domain and thus B�zout domains are GCD domains.
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| 37. | Since fields are integral domains, this is also a construction for the completion of a field with respect to an absolute value.
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| 38. | A Euclidean domain is always a principal ideal domain ( PID ), an integral domain in which every ideal is a principal ideal.
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| 39. | The converse is clear : an integral domain has no nonzero nilpotent elements, and the zero ideal is the unique minimal prime ideal.
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| 40. | If is an integral domain, then by the same reasoning, the fixed points of Frobenius are the elements of the prime field.
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