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The definitions of 'ring' and 'field' are pretty straightforward. For a ring (e.g. integers):

  • addition is commutative $( 1 + 2 = 2 + 1 )$
  • addition and multiplication are associative $(2 +(2+2)) = ((2 + 2) + 2)$
  • multiplication distributes over addition $( 2*(5 + 7) = 2*5 + 2*7 )$
  • each element has an additive inverse $( 2 + -2 = 0 )$
  • there exists an additive identity $( 2 + 0 = 2 )$

The word ring, though, has a concrete meaning in English. Its round. Usually when people name something, it is a metaphor of some kind. Where's the metaphor for 'ring'? What about 'field'?

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As mentioned in the comments, the term 'ring' came from Hilbert (previous question). Bill's answer there is excellent.

Fields are a bit funnier. It started with Dedekind using the word "Zahlenkörper" (body of numbers). In a supplement that he wrote to Dirichlet's Vorlesungenueber Zahlentheorie, he used that word (looking for an image for that) instead of 'rationally known quantities.' It was Moore who coined 'field' - in 1893, he wrote on Galois fields for the Bulletin of the New York Mathematical Society. However, he was always careful to mention 'fields of order' something, as field had the additional meaning at the time of neighborhood (as we understand it today).

I believe he used 'body of numbers' because they stay together just as a group does. But the transition to 'field' is a harder one to understand. Perhaps he meant it as an extension of the neighborhood-field idea - these numbers stayed in a certain neighborhood of a set of numbers in the sense that they were only a couple of powers of s away (referring to his passage below). Or perhaps he meant in to be similar to a 'body', in the sense that they just stay together.

In any case, if there were a strong, direct metaphor for 'ring' or 'field,' it is no longer readily apparent. The origins of the terms are difficult to track and harder to understand completely, and different people used the same terms in different ways. It is even possible that little thought was given to the naming of these things - it is most often hard to judge the importance of something new. Reaching for the term 'body' or 'field' because it fits and doesn't already mean something else might be the real origin.

Here is part of his piece on Galois fields for the Bulletin of the NYMS

enter image description here

Later, Edward Huntington wrote about the development of the theory of fields (it's funny how today we think of fields and groups as being basic, though groups are far older)

Here is an excerpt from Huntington's piece:

Huntington:

enter image description here

enter image description here

with footnote - indicating the well-understood meaning of field today:

enter image description here

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    $\begingroup$ As a sidenote: we call fields cuerpos in Spanish, which means bodies $\endgroup$ – F M Oct 9 '11 at 18:10
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    $\begingroup$ You're not gonna find a better answer to the question then that. Great job,mixed! $\endgroup$ – Mathemagician1234 Oct 9 '11 at 18:16
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    $\begingroup$ As another sidenote: we call fields corps in French, which means bodies $\endgroup$ – Eric Naslund Oct 9 '11 at 18:20
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    $\begingroup$ And it is still "Körper" in German. $\endgroup$ – starblue Oct 9 '11 at 18:45
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    $\begingroup$ @FernandoMartin: In some places, maybe. In Mexico, we called them "Campos" (which means "fields"), not "cuerpos". $\endgroup$ – Arturo Magidin Oct 9 '11 at 19:41

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