p-adic numbers. Part 2: p-adic powers

This is the second 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 define the p-adic integers for p a prime, and use this to construct the field of p-adic numbers. We show how do do various operations, such as taking square roots or powers or exponentials or logarithms with them. 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 3 of talk: https://youtu.be/FWE3mA2otTU 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)