ЛЕКЦИЯ №4 - КОНЕЧНЫЕ ПОЛЯ

"Finite Mathematics" outlines the limits of the applicability of everyday intuition when working with mathematical abstractions. How many points are there in a 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.