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
Proof of Lemma and Lagrange's Theorem
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Proof of Lemma and Lagrange's Theorem. This video starts by proving that any two right cosets have the same cardinality. Then we prove Lagrange's Theorem which says that if H is a subgroup of a finite group G then the order of H div
From playlist Abstract Algebra
Berge's lemma, an animated proof
Berge's lemma is a mathematical theorem in graph theory which states that a matching in a graph is of maximum cardinality if and only if it has no augmenting paths. But what do those terms even mean? And how do we prove Berge's lemma to be true? == CORRECTION: at 7:50, the red text should
From playlist Summer of Math Exposition Youtube Videos
Burnside's Lemma (Part 2) - combining math, science and music
Part 1 (previous video): https://youtu.be/6kfbotHL0fs Orbit-stabilizer theorem: https://youtu.be/BfgMdi0OkPU Burnside's lemma is an interesting result in group theory that helps us count things with symmetries considered, e.g. in some situations, we don't want to count things that can be
From playlist Traditional topics, explained in a new way
This lecture is part of an online course on rings and modules. We continue the previous lecture on complete rings by discussing Hensel's lemma for finding roots of polynomials over p-adic rings or over power series rings. We sketch two proofs, by slowly improving a root one digit at a tim
From playlist Rings and modules
FIT3.1.4. Factoring Example: Artin-Schreier Polynomials
Field Theory: We show that g(x)=x^5-x+1 is irreducible over the rationals using techniques from finite fields. This leads to the definition of an Artin-Schreier polynomial, and in turn we obtain a class of irreducible polynomials over the rationals and prime characteristic.
From playlist Abstract Algebra
Moving on from Lagrange's equation, I show you how to derive Hamilton's equation.
From playlist Physics ONE
DATE & TIME 05 November 2016 to 14 November 2016 VENUE Ramanujan Lecture Hall, ICTS Bangalore Computational techniques are of great help in dealing with substantial, otherwise intractable examples, possibly leading to further structural insights and the detection of patterns in many abstra
From playlist Group Theory and Computational Methods
Alexander HULPKE - Computational group theory, cohomology of groups and topological methods 1
The lecture series will give an introduction to the computer algebra system GAP, focussing on calculations involving cohomology. We will describe the mathematics underlying the algorithms, and how to use them within GAP. Alexander Hulpke's lectures will being with some general computation
From playlist École d'Été 2022 - Cohomology Geometry and Explicit Number Theory
Galois theory: Artin Schreier extensions
This lecture is part of an online graduate course on Galois theory. We describe the Galois extensions with Galois group cyclic of order p, where p is the characteristic of the field. We show that these are generated by the roots of Artin-Schreier polynomials. Minor correction: Srikanth
From playlist Galois theory
Lyapunov Stability via Sperner's Lemma
We go on whistle stop tour of one of the most fundamental tools from control theory: the Lyapunov function. But with a twist from combinatorics and topology. For more on Sperner's Lemma, including a simple derivation, please see the following wonderful video, which was my main source of i
From playlist Summer of Math Exposition Youtube Videos
How Small Can a Group or a Graph be to Admit a Non-Trivial Poisson Boundary? - Anna Erschler
Group Theory/Dynamics Talk Topic: How Small Can a Group or a Graph be to Admit a Non-Trivial Poisson Boundary? Speaker: Anna Erschler Affiliation: École normale supérieure Date: December 9, 2022 We review results about random walks on groups, discussing results and conjectures relating c
From playlist Mathematics
Theory of numbers: Gauss's lemma
This lecture is part of an online undergraduate course on the theory of numbers. We describe Gauss's lemma which gives a useful criterion for whether a number n is a quadratic residue of a prime p. We work it out explicitly for n = -1, 2 and 3, and as an application prove some cases of Di
From playlist Theory of numbers
The dynamics of Aut(Fn) actions on group presentations and representations - Alexander Lubotzky
Character Varieties, Dynamics and Arithmetic Topic: The dynamics of Aut(Fn) actions on group presentations and representations Speaker: Alexander Lubotzky Affiliation: Hebrew University of Jerusalem; Visiting Professor, School of Mathematics Date: December 15, 2021 Several different area
From playlist Mathematics
Building Expanders in Three Steps - Amir Yehudayoff
Amir Yehudayoff Technion-Israel; Institute for Advanced Study February 23, 2012 The talk will have 2 parts (between the parts we will have a break). In the first part, we will discuss two options for using groups to construct expander graphs (Cayley graphs and Schreier diagrams). Specifica
From playlist Mathematics
Galois theory for Schrier graphs: bounded automata by Hemant Bhate
PROGRAM DYNAMICS OF COMPLEX SYSTEMS 2018 ORGANIZERS Amit Apte, Soumitro Banerjee, Pranay Goel, Partha Guha, Neelima Gupte, Govindan Rangarajan and Somdatta Sinha DATE: 16 June 2018 to 30 June 2018 VENUE: Ramanujan hall for Summer School held from 16 - 25 June, 2018; Madhava hall for W
From playlist Dynamics of Complex systems 2018
Proof & Explanation: Gauss's Lemma in Number Theory
Euler's criterion: https://youtu.be/2IBPOI43jek One common proof of quadratic reciprocity uses Gauss's lemma. To understand Gauss's lemma, here we prove how it works using Euler's criterion and the Legendre symbol. Quadratic Residues playlist: https://www.youtube.com/playlist?list=PLug5Z
From playlist Quadratic Residues
Steven Galbraith, Isogeny graphs, computational problems, and applications to cryptography
VaNTAGe Seminar, September 20, 2022 License: CC-BY-NC-SA Some of the papers mentioned in this talk: Ducas, Pierrot 2019: https://link.springer.com/article/10.1007/s10623-018- 0573-3 (https://rdcu.be/cVYrC) Kohel 1996: http://iml.univ-mrs.fr/~kohel/pub/thesis.pdf Fouquet, Morain 2002: ht
From playlist New developments in isogeny-based cryptography
Henry Adams (8/30/21): Vietoris-Rips complexes of hypercube graphs
Questions about Vietoris-Rips complexes of hypercube graphs arise naturally from problems in genetic recombination, and also from Kunneth formulas for persistent homology with the sum metric. We describe the homotopy types of Vietoris-Rips complexes of hypercube graphs at small scale param
From playlist Beyond TDA - Persistent functions and its applications in data sciences, 2021
In this video, I prove the famous Riemann-Lebesgue lemma, which states that the Fourier transform of an integrable function must go to 0 as |z| goes to infinity. This is one of the results where the proof is more important than the theorem, because it's a very classical Lebesgue integral
From playlist Real Analysis