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The discrepancy regarding the names of commutative division algebras in German and English has always startled me. In English they are called fields, whereas their original German name is Körper (hence the $K$), a word which usually means "body" in everyday language. Nearly all other English names of algebraic objects are direct translations of the corresponding German terms (or maybe also the other way round), so I'm wondering which historic development lead to this divergence in naming conventions.

By looking up the Wikipedia article on fields in the respective languages, I found out that most languages seem to have adopted the German name. For example a field is called corps commutatif in French, cuerpo in Spanish, Σώμα in Greek, corpus in Latin, Ciało in Polish, kropp in Swedish and Norwegian.

However there also seem to be some languages where the name for fields translates to "field", such as Russian (Поле) and Italian (campo). Dutch seems to assume a special role, since the term used in the Netherlands seems to be lichaam, whereas the Belgians call it veld.

Does anyone know how this strange division in naming conventions did evolve?

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I do not speak dutch, but wiktionary: en.wiktionary.org/wiki/lichaam translates the everyday meaning of lichaam with body. –  Lennart Meier Jan 22 '14 at 22:25
@LennartMeier: And veld translates to field, so the Dutch use the "German" and the Dutch-speaking Belgians the "English" term. –  Dominik Jan 22 '14 at 22:31
Curiously, in Russian, a commutative field sounds as field (поле), and a skew field sounds as body (тело). Sort of mixture. –  Sasha Anan'in Jan 22 '14 at 22:33
@SashaAnan'in: the same happens in Italian, where the word for skew field is "corpo" (literally, "body"). Also, in French "corps" denotes a skew field. –  Marco Golla Jan 22 '14 at 22:50
Marco: in Flemish Dutch, the same happens: they use the word lichaam (body) for skew field and, as noted above, veld (field) for field. –  René Jan 23 '14 at 2:08

3 Answers 3

up vote 15 down vote accepted

It might be worth noting that the word "Feld" is also (sometimes) used in German, at least in the compound "Galoisfeld" (for "Galois field"). And, this is not some modern-day re-translation from English but classical usage (of Witt for example), see the discussion in comments of http://mathoverflow.net/a/18638/9072 for references.

In my opinion this provides additional support for anon's answer that quite at the start there were two different names and just in the one language the one 'won' whereas in the other one the other 'won' the 'competition'.

Regarding other theories and in general, it might be worth noting that "Körper" does not only mean body, like in human-body, but is also used to designate certain structures/organizations/groups, like "Justizkörper" (comprising judges, attorneys, and so on). And, in that meaning "Körper" fits a lot better with "Gruppe" (group) and also with "Ring" (also used as name for organizations, like in English) and also "Verband" (again used to designate organizations, the math meaning being lattice, in the order sense, providing another example where it is not a translation). See http://mathoverflow.net/a/117354/9072 for some discussion. So that "Körper" is really not all that sensual in that context but quite corporate (which ultimately derives from the same source, I think).

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The word 'body' has a similar alternative meaning in English (see meaning 13a in the OED). –  HJRW Jan 23 '14 at 13:26
One of the answers at mathoverflow.net/questions/35286/… points to some examples in the early 20th century where the terms "body" and "realm" were used in place of "field" in English. –  KConrad Jan 23 '14 at 13:45
As far as I see, the circle around Witt, F. K. Schmidt, Hasse etc. used "Körper" in the modern meaning; the only usage of "Feld" was in the compound "Galoisfeld" for a finite field. E.g. on the last page of Witt's "Riemann-Rochscher Satz und Z-Funktion im Hyperkomplexen" he talks about an "algebraischer Funktionenkörper mit einem Galoisfeld als Konstantenkörper", that is an "algebraic function field (Körper) having a Galois field (Galoisfeld) as field (Körper) of constants". –  Torsten Schoeneberg Jan 23 '14 at 14:20
In French too, the word "corps" can be used in this second sense. For instance, civil servants with a permanent position are organized in corps (corps des agrégés, corps des ingénieurs de l'armement, and so on). –  Denis Serre Jan 23 at 8:46

I think the explanation is that the concept arose somewhat independently with English-speaking mathematicians. See the discussion of the earliest known use at http://jeff560.tripod.com/f.html

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Interesting. Then, apparently the Chinese imported math from English speaking mathematicians (field = 域) while the Japanese did from German speakers (körper = 体). –  Yuichiro Fujiwara Jan 23 '14 at 3:49
@YuichiroFujiwara Chinese 体 refers to skew field. –  Ma Ming Jan 23 '14 at 13:56
@YuichiroFujiwara Takagi (at the turn of the century essentially the only Japanese research mathematician, as far as I know) wrote his Ph.D. thesis in German, after studying at Göttingen under the supervision of Hilbert. German had a lot of currency as a mathematical language before the First World War. –  Ben Webster Jan 23 '14 at 21:03

I once heard it was because "body" is too sensual for (Victorian?) England, so "field" was preferred. As @HJRW pointed out, this may be made up. The circumstantial evidence seems to be there, though:


The Victorian era of British history was the period of Queen Victoria's reign from 20 June 1837 until her death, on 22 January 1901.

Prospect Magazine:

By the Victorian era the body had become so offensive that innocuous words like “leg” were being euphemised as “limb” or “lower extremity.

Link suggested in anon's answer:

Eliakim Hastings Moore (1862-1932) was apparently the first person to use the English word field in its modern sense and the first to allow for a finite field. He coined the expressions "field of order s" and "Galois-field of order s = qn." These expressions appeared in print in December 1893 in the Bulletin of the New York Mathematical Society III. 75.

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This sounds made up. I think a source is needed. –  HJRW Jan 23 '14 at 6:29
Although there is of course some truth to the stereotype of the prudish Victorians, it's almost always overstated. –  HJRW Jan 23 '14 at 9:36
I fear the Prospect magazine article overstates the case. The OED lists many polite Victorian uses of the word 'trousers' (by Charles Dickens and George Eliott, among others). I also fear that the author has swallowed a well known canard. Wikipedia says: 'there is a myth, started by Frances Trollope's Domestic Manners of the Americans, and later applied to the British, that furniture such as tables were covered with embroidery and tablecloths so that table legs were hidden from view, but no historical evidence suggest that this was actually practiced.' –  HJRW Jan 23 '14 at 12:14
E.H. Moore was American and, to add another curious twist, he studied in Germany! –  Michael Renardy Jan 23 '14 at 13:14
I heard that an English translation of an early German book in algebra used "body expansion" for a field extension. I don't know if it is true. –  P Vanchinathan Jan 23 at 8:47

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