p adic numbers part 3: the p-adic gamma function
Channel: Richard E Borcherds
Date: 2020-09-04

This is the third part of a 3-part talk on p-adic numbers for advanced high school students. It is part of a series organized by the Berkeley mathematics circle.

We explain how to define a sort of p-adic analog of the factorial function by modifying the usualy definition of the factorial function. This gives a sort of factorial modulo any prime power, and these can be joined together to form a p-adic factorial. The construction uses Wilson's theorem, which we explain. 

Links related to the video:
Berkeley math circle: https://mathcircle.berkeley.edu/
Handout for talk: https://drive.google.com/file/d/1-UUjAqOIPK-PWUElAXWTdXJZVq7LFGAe
Part 1 of talk: https://youtu.be/VTtBDSWR1Ac
Part 2 of talk: https://youtu.be/PeM5Hp0gWf4

Further reading:
Borevich and Shafarevich, Number theory, chapter 1.3 (advanced)
J.-P. Serre, A course in arithmetic, chapter II (more advanced)
N. Koblitz, p-adic numbers, p-adic functions, and zeta functions (very advanced, for anyone who is really ambitious)