Many thanks for your response.
>
> I have a xerox-copy of Moore's paper. The footnote reads:
>
> Note: This paper was the subject of an address to the Congress
> August 25, 1893. In preparing it for publication the list of
> \S 1 has been revised, the last paragraph of \S 1 added, and
> the details of the proof of the theorem of \S 3 inserted.
> The term "field" (\S 3) and W e b e r ' s term "endlicher K"orper"
> are synonyms. W e b e r , "Die allgemeinen Grundlagen der
> Galois'schen Gleichungstheorie" ("Mathematische Annalen," vol.
> 43, pp. 521-549, November, 1893).
>
> What I set in quotation marks is in fact written in italics in the
> original and \S is replacing the paragraph sign which is not on
> my key board.
>
> As there is no indication that the editor added the footnote, I believe
> that the footnote is Moore's despite of its being formulated very
> neutrally.
And despite the neutral wording, I am not at all convinced that the
footnote was actually added by Moore himself. It might be a trivial
concern of my part but, as I already stated, Moore coined the
expressions
"field of order s"
and
"Galois-field of order s=q^n"
and, as far as I am aware of, (in the early nineties) he basically did not
refer to FIELD or GALOIS-FIELD alone.
According to the abstract of a paper - that I have in front of me - presented
to the Congress of Mathematics at Chicago, August 25, 1893, Moore wrote:
"3. _Galois-field of order $s$ = $q^n$_
Suppose we have a system of symbols or _marks_, \mu_1, \mu_2 ...
\mu_s, in numbers $s$, and suppose that these $s$ marks may be
combined by the four fundamental operations of algebra ... Such a
system of $s$ marks we call a _field of order $s$_.
The most familiar instance of such a field, of order $s$ = $q$ = $a$
prime, is the system of $q$ incongruous classes (modulo $q$) of
rational integral numbers $a$.
...
It should be remarked further that every field of order $s$ is in
fact abstractly considered a Galois-field of order $s$ = $q^n$."
[ A side-note to Jeff Miller. It is under these circumstances that Moore
introduced the symbol GF[q^n], which is, slightly changed, the one we use
today for "Galois-field of order q^n". ]
Thus, the wording of the footnote
"The term "field" and W e b e r ' s term "endlicher K"orper"
are synonyms. W e b e r, "Die allgemeinen Grundlagen der
Galois'schen Gleichungstheorie" ("Mathematische Annalen",
vol. 43, pp. 521-549, November, 1893).
sounds a bit odd. Hastings Moore seems to have been very careful in writing,
and in systematically referring to
FIELD of ORDER $s$ ,
and not merely to the (shorter and more economic) term
FIELD.
But... why "field of order $s$" instead of "field"? Professor John Harper
may be right when he remarks that:
At that time "field" seems to have had a different mathematical
meaning from its present one, at least for some authors. In Harkness,
J. and Morley, F. 1893, A treatise on the theory of functions, G.E.
Stechert & Co., New York, the word seems to mean "interval". On p.49
(of the 1925 reprint; I do not know if the page number was the same
in 1893):
"The function f(x) is said to be continuous at the point c ...
if a field (c-h to c+h) can be found such that for all points
of this field, |f(x)-f(c)| < epsilon."
I looked for a definition on and before p.49 but failed to find one;
that suggests to me that Harkness and Morley thought it was common
usage.
The previous sense of "field" as "neighbourhood" may have guided Moore's
choice. Obviously enough, a term used in analysis would not be confused
with a term thought to play its game on algebraic fields (pun intended).
Anyhow, the idea (and acknowledgement) that the term "field" and Weber's
term "endlicher Ko"rper were synonyms *seems* to sail other waters than
those in which the term "field" almost immediately found its path on new
fields.
At any events, a decade later Edward V. Huntington wrote: "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 _Ko"rper_;
KRONECKER's term _Rationalita"tsbereich_, 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, WEBER's
definition of an abstract field being substantially as follows:
[...]
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."
The issue that the term "field", at the turn of the century, already had
the same mathematical meaning as its present one makes it clear from the
following footnote:
"The most familiar and important example of an infinite field is
furnished by the rational numbers, under the operations of ordinary
addition and multiplication. In fact, a field may be briefly described
as a system in which the rational operations of algebra may all be
performed (excluding division by zero). A field may be finite, provided
the number of elements (called the order of the field) is a prime or a
power of a prime." [Paper by EVH presented to the AMS on December 30,
1904, and received for publication on February 9, 1905.]
Best regards,
Julio Gonzalez Cabillon