| 词汇 | unique-factorization-domain |
| 释义 | BETA unique factorization domaincollocation in Englishmeanings of unique, factorizationand domainThese words are often used together. Click on the links below to explore the meanings. unique adjective uk /juːˈniːk/ us /juːˈniːk/ being the only existing one of its type or, more generally, unusual, or special in ... See more at unique factorization noun[ C or U ] uk /ˌfæk.tər.aɪˈzeɪ.ʃən/ us /ˌfæk.tɚ.əˈzeɪ.ʃən/ the dividing of a number into factors (= whole numbers that are produced when you divide a larger number by another ... See more at factorization domain noun[ C ] uk /dəˈmeɪn/ us /doʊˈmeɪn/ an area of interest or an area over which a person ... See more at domain Examples of unique factorization domainunique factorization domain Since a uniquefactorizationdomain is precisely a ring in which an analogue of the fundamental theorem of arithmetic holds, this question is readily answered. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. This ring is a frequently cited example of an integral domain that is not a uniquefactorizationdomain. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. As for every principal ideal domain, is also a uniquefactorizationdomain. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. A uniquefactorizationdomain is not necessarily a noetherian ring. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. Thus, a number field has class number 1 if and only if its ring of integers is a principal ideal domain (and thus a uniquefactorizationdomain). From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The greatest common divisor may be defined and exists, more generally, for multivariate polynomials over a field or the ring of integers, and also over a uniquefactorizationdomain. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. This property of unique factorization is commonly expressed by saying that the polynomial rings over a field or a uniquefactorizationdomain are unique factorization domains. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. A third reason is that the theory and the algorithms for the multivariate case and for coefficients in a uniquefactorizationdomain are strongly based on this particular case. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The rings where every irreducible is prime are called unique factorization domains. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The converse does not hold in general, but does hold for unique factorization domains. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. The fundamental theorem of arithmetic continues to hold in unique factorization domains. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. This led to the study of unique factorization domains, which generalize what was just illustrated in the integers. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. They showed that regular local rings of dimension 3 are unique factorization domains, and had previously shown that this implies that all regular local rings are unique factorization domains. From Wikipedia This example is from Wikipedia and may be reused under a CC BY-SA license. These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors. |
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