Abel's irreducibility theorem

In mathematics, Abel's irreducibility theorem, a field theory result described in 1829 by Niels Henrik Abel, asserts that if ƒ(x) is a polynomial over a field F that shares a root with a polynomial g(x) that is irreducible over F, then every root of g(x) is a root of ƒ(x). Equivalently, if ƒ(x) shares at least one root with g(x) then ƒ is divisible evenly by g(x), meaning that ƒ(x) can be factored as g(x)h(x) with h(x) also having coefficients in F. Corollaries of the theorem include: * If ƒ(x) is irreducible, there is no lower-degree polynomial (other than the zero polynomial) that shares any root with it. For example, x2 − 2 is irreducible over the rational numbers and has as a root; hence there is no linear or constant polynomial over the rationals having as a root. Furthermore, there is no same-degree polynomial that shares any roots with ƒ(x), other than constant multiples of ƒ(x). * If ƒ(x) ≠ g(x) are two different irreducible monic polynomials, then they share no roots. (Wikipedia).

Differential Equations | Abel's Theorem

We present Abel's Theorem with a proof. http://www.michael-penn.net

From playlist Differential Equations

Abel formula

This is one of my all-time favorite differential equation videos!!! :D Here I'm actually using the Wronskian to actually find a nontrivial solution to a second-order differential equation. This is amazing because it brings the concept of the Wronskian back to life! And as they say, you won

From playlist Differential equations

Irreducibility and the Schoenemann-Eisenstein criterion | Famous Math Probs 20b | N J Wildberger

In the context of defining and computing the cyclotomic polynumbers (or polynomials), we consider irreducibility. Gauss's lemma connects irreducibility over the integers to irreducibility over the rational numbers. Then we describe T. Schoenemann's irreducibility criterion, which uses some

From playlist Famous Math Problems

Irreducibility (Eisenstein's Irreducibility Criterion)

Given a polynomial with integer coefficients, we can determine whether it's irreducible over the rationals using Eisenstein's Irreducibility Criterion. Unlike some our other technique, this works for polynomials of high degree! The tradeoff is that it works over the rationals, but need not

From playlist Modern Algebra - Chapter 11

Henri Darmon: Andrew Wiles' marvelous proof

Abstract: Pierre de Fermat famously claimed to have discovered “a truly marvelous proof” of his last theorem, which the margin in his copy of Diophantus' Arithmetica was too narrow to contain. Fermat's proof (if it ever existed!) is probably lost to posterity forever, while Andrew Wiles' p

From playlist Abel Lectures

Differential Equations | Application of Abel's Theorem Example 1

We give an example of applying Abel's Theorem to construct a second solution to a differential equation given one solution. www.michael-penn.net

From playlist Differential Equations

Differential Equations | Application of Abel's Theorem Example 2

We give an example of applying Abel's Theorem to construct a second solution to a differential equation given one solution. www.michael-penn.net

From playlist Differential Equations

SHM - 16/01/15 - Constructivismes en mathématiques - Frédéric Brechenmacher

Frédéric Brechenmacher (LinX, École polytechnique), « Effectivité et généralité dans la construction des grandeurs algébriques de Kronecker »

Math 139 Fourier Analysis Lecture 07: Cesaro and Abel summability

Cesaro and Abel Summability: Cesaro mean (Cesaro sum); Fejer kernel; closed form of the Fejer kernel; Fejer kernels are good kernels (approximations of the identity); consequences. Abel means; Abel summable. Abel means of the Fourier series of a function. Abel means as convolution with

From playlist Course 8: Fourier Analysis

Galois theory: Abel's theorem

This lecture is part of an online graduate course on Galois theory. We discuss Abel's theorem, that says a general quintic equation cannot be solved by radicals. We do this by showing that if a polynomial can be solved by radicals over a field of characteristic 0 then its roots lie in a s

From playlist Galois theory

Bourbaki - 16/01/2016 - 4/4 - Benoît STROH

Benoît STROH La correspondance de Langlands sur les corps de fonctions, d’après V. Lafforgue La moitié de la correspondance de Langlands sur les corps de fonctions prédit qu’à toute représentation automorphe des points adéliques d’un groupe G on peut associer un système local sur un ouvert

From playlist Bourbaki - 16 janvier 2016

Abel Prize — The story

A short introduction to the story of The Abel Prize, from the story of Niels Henrik Abel, the establishment of the prize in 2003, and the way mathematics impact our everyday lives. The main objective of the Abel Prize is to recognize pioneering scientific achievements in mathematics. The

From playlist About the Abel Prize

Why you can't solve quintic equations (Galois theory approach) #SoME2

An entry to #SoME2. It is a famous theorem (called Abel-Ruffini theorem) that there is no quintic formula, or quintic equations are not solvable; but very likely you are not told the exact reason why. Here is how traditionally we knew that such a formula cannot exist, using Galois theory.

From playlist Traditional topics, explained in a new way

Michel Broué: Building Cathedrals and breaking down Reinforced Concrete Walls

This lecture was held at The University of Oslo, May 21, 2008 and was part of the Abel Prize Lectures in connection with the Abel Prize Week celebrations. Program for the Abel Lectures 2008 1. Abel Laureate John Thompson: “Dirichlet series and SL(2,Z)" 2. Abel Laureate Jacques Tits: “Alg

From playlist Abel Lectures

The Abel lectures: Hillel Furstenberg and Gregory Margulis

0:30 Welcome by Hans Petter Graver, President of the Norwegian Academy of Science Letters 01:37 Introduction by Hans Munthe-Kaas, Chair of the Abel Prize Committee 04:16 Hillel Furstenberg: Random walks in non-euclidean space and the Poisson boundary of a group 58:40 Questions and answers

From playlist Gregory Margulis

Pierre Colmez - Sur le programme de Langlands p-adique

Le programme de Langlands p-adique a pour origine les travaux de Serre et de Hida sur les familles p-adiques de formes modulaires et les représentations galoisiennes qui leur sont associées. Mazur, en collaboration avec Gouvéa et avec Coleman, a joué un grand rôle dans la maturation de ce

From playlist Journée Gretchen & Barry Mazur

Umberto Zannier - Unlikely Intersections and Pell's equations in polynomials

Unlikely Intersections and Pell's equations in polynomials

From playlist 28ème Journées Arithmétiques 2013

Omer Offen: Period integrals of automorphic forms

Recording during the thematic Jean-Morlet Chair - Doctoral school: "Introduction to relative aspects in representation theory, Langlands functoriality and automorphic forms" the May 18, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume H

Introduction to additive combinatorics lecture 10.8 --- A weak form of Freiman's theorem

In this short video I explain how the proof of Freiman's theorem for subsets of Z differs from the proof given earlier for subsets of F_p^N. The answer is not very much: the main differences are due to the fact that cyclic groups of prime order do not have lots of subgroups, so one has to