I'm interested in learning about the historical motivation and development of $p$-adic numbers. I haven't been able to find any books on the topic. I'd appreciate any references, including to more general history books which include coverage of the $p$-adics. Alternatively, if anyone has any knowledge about the history of $p$-adic numbers, feel free to post a summary here, particularly if you can highlight any names, papers and keywords that I could use to do more research on my own.
I'm not looking for a simplified overview, I want to really dig into the details, but any amount of information that could get me started is appreciated.
Best Answer
One useful starting point is Peter Roquette's History of Valuation Theory. This may be a bit more general than you seek, but it has a bibliography that should provel helpful locating other sources. See also F. Gouvea, Hensel’s p-adic Numbers: early history, and the following
Ullrich, Peter, 1998: The genesis of Hensels p-adic numbers.
In the book: $ $ P. L. Butzer, H.Th. Jongen,W. Oberschelp (edd.), Charlemagne and his Heritage. 1200 Years of Civilization and Science in Europe. / Karl der Große und sein Nachwirken. 1200 Jahre Kultur und Wissenschaft in Europa. Vol. 2. Mathematical Arts. Thurnhout: Brepols, 163-178