(#008) p-adics - Siddharth Bhat
Date & Time: 10-10-2020, 23:00 IST
Abstract
The integers and polynomials of a single variable look tantalizingly similar. Many of our intuitions about factorizations, primality, GCD, LCM, and other number-theoretic operations work "equally well" on single-variable polynomials.
However, single-variable polynomials appear to provide a richer theory at first glance: We can evaluate polynomials, differentiate them, take their taylor series expansion, and other operations which are hinged on the ability to interpret a polynomial as a function. We shall embark on a quest to port these operations back into number-land. This will lead us to eventually define the p-adics, which can be viewed as the "correct way" to transport the ability to evaluate a single-variable polynomial back into the integers.
Some cute counter intuitive results will be shown. Delight shall hopefully be had. No theorems of consequence will be demonstrated, since my knowledge of these p-adics is limited.
Resources
The reference material for the talk is the first three to four chapters of the book, "p-adic numbers, an introduction" by Fernando Gouvea