Лекция №2 — КОНЕЧНЫЕ ПОЛЯ

FINITE FIELDS "Finite mathematics" outlines the limits of the applicability of everyday intuition when working with mathematical abstractions. How many points are there in the plane? How many fifth-degree polynomials are there? How many times must one add one to itself to get zero? These seemingly absurd questions are a prelude to the material of our four-lecture mini-course. Here is a rough outline: 1. Tables of addition and multiplication of remainders. Polynomials with coefficients in remainders. Bézout's theorem over any system of remainders. Paradoxes of the number of roots. 2. Multiplication tables modulo a prime. The simplest finite fields. Fundamental theorem on the roots of polynomials with coefficients in a field. 3. Fields of p elements (p is a prime number). Group-theoretic methods: Lagrange's theorem and Fermat's Little Theorem. Newton's binomial theorem, the p-th power automorphism, and a second proof of Fermat's theorem. Wilson's theorem. 4. Finite fields with p^r elements, the multiplicative group, and the structure of their embeddability. Uniqueness of a finite field. ➖➖➖➖➖➖➖ 💪🏻Support our project, become our patron:   / savvateev   Our resources: https://vk.com/alexei_savvateev   / aleksey_savvateev     / savvatan   https://savvateev.livejournal.com https://savvateev.xyz https://t.me/savvateev_xyz https://t.me/punkmath 📚Alexey Savvateev's book "Mathematics for Humanities Students": https://www.savvateev.xyz/ Project team: Valery Dragun Eduard Dubnitsky Pavel Ivanov Nikolay Kazimirov Egor Kuzmichev Kirill Kuchin Alexey Savvateev Daria Fedorova ❗️Thank you to Igor Gitman for your help And a special thanks to our Patrons (patreon.com/savvateev), who make high-quality studio recordings and many other improvements to the channel possible.