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I am interested in getting the original information regarding Dedekind's idea of the "different" (regarding ideals). Particularly, I am interested in:

1- Knowing the original German name he used for it. I think it was "diskriminant", but I am not sure. 2-Getting his original definition (It would be great if the page could be scanned and bibliographic reference could be given).

Thanks."

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See :

in 1882, Dedekind completes the theory [of ideals] by introducing the different, which gives him a new definition of the discriminant and allows him to specify the exponents of the prime ideal factors in the decomposition of the latter.

In the German text of : Richard Dedekind, Über die Discriminanten endlicher Körper, 1882, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 29 (2): 1–56, see page 1 for : "Grundzahl oder Discriminante" and page 38 for : "Grundideal des Körpers".

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  • $\begingroup$ It is still not clear to me whether the original term fuer the different was discriminant from the very beginning...... $\endgroup$
    – Javier Arias
    Commented Sep 16, 2015 at 17:40
  • $\begingroup$ My question would then be: Which term does Dedekind use in his 1882 paper to mention what has ben translated as the different (regarding ideals)? $\endgroup$
    – Javier Arias
    Commented Sep 16, 2015 at 18:47
  • $\begingroup$ So das Wort war dis discriminante? Wikipedia refers rather to the 1882 paper Cf. Richard Dedekind, (1882), “Über die Discriminanten endlicher Körper”, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 29, 2, 1882, p. 1–56. Is that a confusion?? $\endgroup$
    – Javier Arias
    Commented Sep 16, 2015 at 18:55
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    $\begingroup$ No, in this 1882 paper, Dedekind calls the discriminant "Discriminante" or "Grundzahl". What we would now call the different (or different ideal) he calls "Grundideal", see p.38. He does not seem to give a name to what we would now call the different of an algebraic element $\theta$, but simply denotes it by $\theta^*$ (defined on p.4). Dedekind would never have used "Discriminante" for the different, because discrimants were well established at that point and are different from 'differents'. $\endgroup$
    – moonlight
    Commented Sep 16, 2015 at 20:58
  • $\begingroup$ The term "Differente" was used by Hilbert in his Zahlbericht (Section 12), and he refers in a footnote to Dedekind having used the term Grundideal, so I suspect this means the modern term was introduced by Hilbert. A link to the relevant page in the Zahlbericht is digizeitschriften.de/dms/img/… (see also the previous page). $\endgroup$
    – KCd
    Commented Apr 25, 2022 at 0:42

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