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Topic: Terminology of integers
Replies: 1   Last Post: Aug 28, 2006 9:30 AM

 Colin McLarty Posts: 11 Registered: 3/3/06
Re: Terminology of integers
Posted: Aug 28, 2006 9:30 AM

Thanks to Lee Rudolph. He is right about van der Waerden's usage. And
``quantity'' is a better translation than ``magnitude.''

It seems that in current German algebra the word ``ganz/integer'' is
only an adjective. Writers say ``integer number'' or ``integer
element'' or ``integer quantity'' in a number field or a function
field. In English, in number fields, people say ``integer'' but in
function fields they prefer ``integral function'' and the like. For a
translation I expect to use "integer" as a noun for Noether's ``ganz
Grosse'' in number fields, and since she uses the phrase just once of
function fields I'll translate it the same way.

best, Colin

Lee Rudolph wrote:
> "colin.mclarty@case.edu" <colin.mclarty@case.edu> writes:
>

> >Certainly some people talk about "integers" in an algebraic function
> >field in the ring-theoretic sense -- algebraic functions that satisfy a
> >monic polynomial over the field of rational functions in one variable.
> >But I do not know how common this is.
> >
> >For a translation of Emmy Noether's paper "Abstrakter Aufbau der
> >Idealtheorie" I need a term for "ganzen Grossen" in an algebraic
> >function field in that sense. This is the same term she uses for
> >integers in an algebraic number field. When she wants to specify
> >integers in the basic arithmetic sense, rational integers in algebraic
> >number fields, she uses "ganzen Zahlen."
> >
> >A word-by-word translation would make "ganzen Grossen" into "integer
> >magnitudes." I have never heard anyone in English talk about "integer
> >magnitudes" in algebraic number fields. I think everyone just says
> >"integers." But is this also common in algebraic function fields?

>
> I believe the phrase "integral element of an algebraic function field"
> is reasonably standard. "Integral quantity" might also do (or once
> have done) the trick. If you can find a German-language copy of
> van der Waerden's _Moderne Algebra_, a check of section 101 (vol. 2,
> ch. XIV), which is titled "The Integral Quantities of a Field" in
> the English translation by Benac, would confirm or disconfirm my
> ungrounded but hopeful conjecture that there at least "integral
> quantities" is translating "ganzen Grossen".
>
> Lee Rudolph