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Where did this terminology first appear, and what was the motivation behind using the word "characteristic" to refer to the property that $1 + 1 + ...+1$ a certain number of times gives us the additive identity?

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  • $\begingroup$ Not an answer for this site, but note that if $1 + 1 + \ldots + 1$ ($n$ times) is $0$, then $x + x + \ldots + x$ ($n$ times) is $0$ for every $x$ and (for a field) vice-versa. $\endgroup$ – Jonathan Cast Nov 14 '16 at 21:41
  • $\begingroup$ For example: Was it taken over from German Charakteristik, and if so what did it mean there? $\endgroup$ – Gerald Edgar Nov 15 '16 at 2:01
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The notion and the term were introduced by Steinitz in his seminal paper Algebraische Theorie der Körper (Algebraic Theory of Fields), Crelle's Journal (1910), 167–309. He also defined such concepts as prime field and the transcendence degree there. In his own words, his goal was "to obtain a general overview of all possible types of fields and to determine their relations with each other". According to Roquette's In Memoriam:

"In any abstract field, Steinitz showed that there is a unique smallest sub eld which he called the prime field; this is either in nite (and then isomorphic to the rationals) or its cardinality is a prime number p (and then it is isomorphic to the integers modulo p ). Accordingly he defined the characteristic of a field to be either 0 or p respectively."

To appreciate the paper's significance it helps to quote van der Waerden, whose Modern Algebra text shaped the field:

"In earlier treatises, number fields, and fields of algebraic functions were usually treated in separate chapters, and finite fields in still another chapter. The first to give a unified treatment, starting with an abstract defnition of field, was E. Steinitz in his 1910 paper. In my Chapter 5, called "Korpertheorie", I essentially followed Steinitz..."

Van der Waerden is a bit too generous, the first to give an abstract definition of a field was Weber (whom Steinitz credits) in Die Allgemeinen Grundlagen der Galois'schen Gleichungstheorie, Math. Ann., 43:521-549, 1893. But he focused on Galois theory rather than studying general fields. As for the motivation for "characteristic", it is a pretty common name in mathematics for a simpler object that "characterizes" something more complex, compare to characteristic of a logarithm, characteristic determinant, polynomial, function, etc. Steinitz may also have been influenced by Dedekind's, Frobenius's and Weber's earlier use of "group character" (German Gruppencharakter). According to Shapiro, (quoted from Earliest Known Uses of Some of the Words of Mathematics):

"However, in Dedekind’s edition of Dirichlet’s Vorlesungen ueber Zahlentheorie in 1894, Dedekind included a footnote in which he singled out the notion of "character," defined it explicitly, and denoted it by chi(n). However, he did not give the function a name. Weber’s Lehrbuch der Algebra, II, 1899, defined the function chi(A) as a "Gruppencharakter," and developed some of its elementary properties..."

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