Today, when the term "field" is defined in algebra, it is almost always stipulated that all fields are commutative. However, the author of these lectures says that this has not always been the case:

In older terminology, a field could be non-commutative, i.e., any ring in which each nonzero element has a two-sided multiplicative inverse. We now call such things “division rings” or “division algebras.”

Is this true? If so, why and when did this shift in the definition of "field" occur? What could I read to learn more?

Thank you in advance for your help!

  • $\begingroup$ An example of a division algebra (i.e. non-commutative field) are quaternions. mathworld.wolfram.com/Quaternion.html $\endgroup$ – Martin Vesely Jun 11 at 9:55
  • $\begingroup$ My Dad's PhD thesis was on "A Class of Nilstable Algebras," which IIRC are algebras with a noncommutative operator. And then there are nonassociative, flexible, nilpotent,... quite the assortment! $\endgroup$ – Carl Witthoft Jun 11 at 11:33
  • $\begingroup$ Maybe it's noteworthy that in French, "corps" can more often include the non-commutative case. E.g. Bourbaki's treatise on the subject is explicitly called "Corps Commutatifs". $\endgroup$ – Torsten Schoeneberg Jun 13 at 2:13

Not quite. There was some vagueness in Dedekind's early formulations, but the tendency was to use "Körper" or "field" when the multiplication is commutative from the start. As a curiosity, in Russian general division algebra is called тело, literally "body", which is apparently the translation of German Körper, as opposed to commutative поле, the translation of English "field".

The original use of German Zahlenkörper (literally, number body), or Körper for short, now translated as field, is attributed to Dedekind's Supplement XI to Dirichlet’s Vorlesungenueber Zahlentheorie, §159 (1871) and his also Stetigkeit und irrationale Zahlen (1872) (the latter has been translated into English in Essays on the Theory of Numbers (1901)). Here is the relevant passage from Supplement XI:

"So far, we have only the considered the whole numbers $0, ± 1, ± 2, ± 3, ± 4...$, namely all those numbers that arise from the number $1$ by repeated addition and subtraction; these numbers are reproduced by addition, subtraction and multiplication, or in other words, the sums, differences and products of two whole numbers are again whole numbers. In contrast, the fourth basic operation, the division, leads to the more comprehensive notion of the rational numbers under which name the quotients of any two integers [with non-zero denominator] will be understood; these rational numbers are evidently reproduced by all four basic operations.

In the future, we want to call every system of real or complex numbers that has this fundamental property of reproduction Zahlenkörper or Körper for short; the system $R$ of all rational numbers is one such Körper, and it is the simplest example of one. This Körper $R$ of rational numbers now consists of whole numbers and fractions, i. e. non-integers; the former we want to call rational integers to distinguish them from new integers to be introduced." [my translation and emphasis]

On the one hand, Dedekind does not explicitly require multiplication to be commutative in this "definition", and this would be in line with Galois' definition of a group, which was not required to be Abelian. On the other hand, he appears to be talking about subfields of "real or complex numbers", which are commutative, and all of his examples are fields of algebraic numbers (Gaussian rationals are considered right after the definition). In particular, he considers neither non-commutative division algebras nor finite fields, even though again, his $R$ is not formally required to be infinite.

Systems of hypercomplex numbers were popular around that time, and some of them (quaternions) had non-commutative multiplication and inverses for non-zeros. Whether Dedekind was inclined to call them Körper as well is hard to say. According to Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics, Wildstrom argued that he was. Miller also suggests that the term did not catch on until early 1890-s, in particular, Moore translated it into English as "field" in 1893 (but the aforementioned 1901 translation of Dedekind uses "body" instead). Here is Huntington's surmise from a paper presented to AMS in 1904:

"Closely connected with the theory of groups is the theory of fields, suggested by GALOIS, and due, in concrete form, to DEDEKIND in 1871. The word field is the English equivalent for DEDEKIND’s term Körper; KRONECKER’s term Rationalitätsbereich, which is often used as a synonym, had originally a somewhat different meaning. The earliest expositions of the theory from the general or abstract point of view were given independently by WEBER and by Moore, in 1893 [...] The earliest sets of independent postulates for abstract fields were given in 1903 by Professor Dickson and myself; all these sets were the natural extensions of the sets of independent postulates that had already been given for groups."

Again, if we want "fields" to be a generalization of groups, rather than Abelian groups, it would have to cover what is now called division algebras. However, in the aforementioned Moore's paper of 1893, and in both 1903 papers of Dickson, Definitions of a Field by Independent Postulates and Huntington, Definitions of a Field by Sets of Independent Postulates the commutativity of multiplication is explicitly assumed.

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    $\begingroup$ It may be of interest to say that in Italian as well "non commutative fields" are called "corpo", which is the translation of "K\"orper", while commutative ones are named "campo" which is the translation of "field". $\endgroup$ – Nicola Ciccoli Jun 19 at 8:29

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