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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](http://jeff560.tripod.com/mathword.html). 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...)2011-12-15
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    Thanks for the link to the page.2011-12-15
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    @J.M.: Thanks for adding terminology.2011-12-15
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    Ramification is a fairly exact translation of the German "Verzweigung", and I am pretty sure that the word was used in German first. It could well have been Riemann himself, who first used it, but to verify it you would need to look at the original texts.2011-12-15
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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), 93-196.2011-12-15
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    @Brian: the use of ramification you described comes in the 2nd half of the 20th century, but the question is about when the term first appeared and the answer will be at least some time in the 19th century (work of Riemann, most likely).2011-12-15
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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.2011-12-15
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    In Riemann's collected works, [available here](http://www.archive.org/stream/117719254#page/n75/mode/2up), 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.2011-12-15

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