Talk:Unique factorization

Article Checklist for "Unique factorization"

Workgroup category or categories	Mathematics Workgroup [Categories OK]
Article status	Developing article: beyond a stub, but incomplete
Underlinked article?	No
Basic cleanup done?	Yes
Checklist last edited by	Greg Martin 22:17, 29 April 2007 (CDT)

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could be a good article

Right now this article only discusses unique factorization of integers - and that's definitely the most important sense of the words "unique factorization". However, there are other mathematical objects for which unique factorization is an important concept: polynomial rings and rings of integers in number fields (such as $\mathbb {Z} [i]$) come immediately to mind. The latter concept, which began to be investigated in relation to early attempts to prove Fermat's Last Theorem, eventually led to the definition of ideals in rings. Anyway, consider this an invitation to be bold and expand the current article. - Greg Martin 22:19, 29 April 2007 (CDT)

If I remember right, if you extend the integers by including ${\sqrt {-3}}$ you get a ring which does not have unique prime factorization. --Catherine Woodgold 20:42, 30 April 2007 (CDT)

I'm not sure whether that's true, because as it turns out $\mathbb {Z} [{\sqrt {-3}}]$ is a little less natural a ring than $\mathbb {Z} {\big [}(1+{\sqrt {-3}})/2{\big ]}$. It's definitely true that unique factorization fails in $\mathbb {Z} [{\sqrt {-5}}]$, the canonical example being $2\times 3=(1+{\sqrt {-5}})(1-{\sqrt {-5}})$. - Greg Martin 13:56, 8 May 2007 (CDT)

induction?

Note that the proof currently given is not formally a proof by induction: instead it uses the phrase "Continuing in this way". We should choose one or the other. - Greg Martin 22:21, 29 April 2007 (CDT)

Ambiguity

This first sentence looks ambiguous to me: "In mathematics, the unique factorization theorem, also known as the fundamental theorem of arithmetic states that every integer can be expressed as a product of prime numbers in essentially only one way." It could mean that no integer can be so expressed in (essentially) more than one way. Or, it could mean that every integer can be so expressed in one way and (essentially) only in that one way. I suggest rewording it so that it's very clear which of the two is meant. In other words: is one thing being proven here (that there aren't two significantly different prime factorizations for the same integer), or are two things being proven (what I just said, and also that there is at least one prime factorization for each integer)? I see similar ambiguity on this point on other web pages on this topic. Some might be clear about what they're stating and proving but not clear about which version is labelled the fundamental theorem of arithmetic, for example. Maybe someone could look up a primary source on the origin of the term "fundamental theorem of arithmetic". --Catherine Woodgold 08:00, 7 May 2007 (CDT)