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.
Lee Rudolph wrote: > "email@example.com" <firstname.lastname@example.org> 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