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- Newsgroups: sci.math
- Path: sparky!uunet!gumby!destroyer!ncar!isis.cgd.ucar.edu!steele
- From: steele@isis.cgd.ucar.edu (Alfred Steele)
- Subject: Euclidean Domain
- Message-ID: <1993Jan8.211440.374@ncar.ucar.edu>
- Sender: news@ncar.ucar.edu (USENET Maintenance)
- Organization: Climate and Global Dynamics Division/NCAR, Boulder, CO
- Date: Fri, 8 Jan 1993 21:14:40 GMT
- Lines: 29
-
-
- In article <HAMMOND.93Jan3134111@annemarie.albany.edu>, hammond@csc.albany.edu (
- William F. Hammond) writes:
- |> In article <Jan.3.02.05.44.1993.24643@spade.rutgers.edu>
- |> cadet@spade.rutgers.edu (Uniquely TiJean) writes:
- [....]
-
- |>If someone can answer the question:
- |>Why there are not other candidates for the Euclidean function?
-
- I have asked the very same question and do not know the answer. The previous
- posters have made reference to Hungerford's Algebra Text
- Hungerford's definition does not use the norm - this is true of most
- texts. His problem #8 Chapter III section 3 (in my edition) asks to show
- a + b(1 + sqrt(19)) a,b \in Q is a principal ideal domain but not a
- Euclidean domain. My guess is that any function that satisfies Hungerford's
- definition of Euclidean domain:
- f : R - {0} \arrow N
- a) a,b \in R and ab \notequal 0 then f(a) \le f(ab)
- b) usual definition with division and f(r) \lt f(b) or r = 0 in
- a = qb + r
- will be equivalent to a norm. Something like this is Ostrowski's theorem
- which states that non-trivial norms (or valuations) are either absolute
- values or p-adic norms.
-
- Any information netters know about this, I would be interested in hearing
- about. What is the "folklore" on the subject?
- Alfred T. Steele
- steele@isis.cgd.ucar.edu
-