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In my studies so far, I have had the word 'ramification' come up in Algebraic Number Theory and Complex Analysis.

The Wikipedia article tells me that 'ramification' is also used in some other fields.

I was wondering when the term 'ramification' was first used in literature, and also the field it was first used in.

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Unfortunately, ‘ramification’ does not seem to appear on Jeff Miller's pages. I believe it was originally used in the theory of Riemann surfaces, and its use spread elsewhere by analogy. (The ring of integers of an algebraic number field, after all, is a Dedekind domain, and so akin to a smooth affine algebraic curve...) –  Zhen Lin Dec 15 '11 at 5:02
    
Thanks for the link to the page. –  Rankeya Dec 15 '11 at 5:04
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It’s also used in infinite partition calculus and thence in set-theoretic topology: roughly speaking, a ramification argument shows the existence of an object by building an infinite tree whose branches are better approximations to it the higher they reach and showing that there is a branch that hits every non-empty level and so is the desired object. In this context the name goes back at least to P. Erdős, A. Hajnal, & R. Rado, Partition relations for cardinal numbers, Acta Math. Acad. Sci. Hung. 16 (1965), 93-196. –  Brian M. Scott Dec 15 '11 at 5:45
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@KCd: I realize that. I was not suggesting that this use was earlier; I was merely adding to the list of fields in which the term is used, in case that turned out also to be of interest for the OP. –  Brian M. Scott Dec 15 '11 at 7:07
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In Riemann's collected works, available here, the word Verzweigung seems to appear first in his Beiträge zur Theorie der durch die Gauss'sche Reihe $F(\alpha,\beta,\gamma,x)$ darstellbaren Functionen (1854), last paragraph of the introduction. –  t.b. Dec 15 '11 at 9:19
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