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
Commutative algebra 51: Hensel's lemma continued
This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. This lecture continues the discussion of Hensel's lemma. We first use it to find the structure of the group of units of the p-
From playlist Commutative algebra
Commutative algebra 50: Hensel's lemma
This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. We describe Hensel's lemma for finding roots of polynomials over complete rings, and give some examples of using it to find wh
From playlist Commutative algebra
Number Theory | Hensel's Lemma
We prove Hensel's Lemma, which is related to finding solutions to polynomial congruences modulo powers of primes. http://www.michael-penn.net Thumbnail Image: By Unknown - Universität Marburg, Public Domain, https://commons.wikimedia.org/w/index.php?curid=9378696
From playlist Number Theory
Title: A Differential Hensel's Lemma for Local Algebras
From playlist Spring 2015
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
The Frobenius Problem - Method for Finding the Frobenius Number of Two Numbers
Goes over how to find the Frobenius Number of two Numbers.
From playlist ℕumber Theory
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
Title: A Differential Hensel's Lemma for Local Algebras
From playlist Spring 2015
Hensel's Lemma Number Theory 15
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From playlist Number Theory
Hensel's Lemma -- Number Theory 15
Suggest a problem: https://forms.gle/ea7Pw7HcKePGB4my5 Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Patreon: https://www.patreon.com/michaelpennmath Merch: https://teespring.com/stores/michael-penn-math Personal Website: http://www.michael-penn.net Randolp
From playlist Number Theory v2
Algebraic geometry 41: Completions
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It reviews completions of rings and Hensel's lemma, and gives an example of two analytically isomorphic singularities.
From playlist Algebraic geometry I: Varieties
Introduction to number theory lecture 19. Hensel and Newton's method
This lecture is part of my Berkeley math 115 course "Introduction to number theory" For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj53L8sMbzIhhXSAOpuZ1Fov8 We describe a method due to Hensel and Newton for lifting a solution of an equation mod p t
From playlist Introduction to number theory (Berkeley Math 115)
Introduction to number theory lecture 20. p-adic numbers.
This lecture is part of my Berkeley math 115 course "Introduction to number theory" For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj53L8sMbzIhhXSAOpuZ1Fov8 We descibe the general case of Hensel's lemma (or Newton's method) for solving equations, a
From playlist Introduction to number theory (Berkeley Math 115)
[ANT15] p-adic integers: a primer, and an application (part 1)
The p-adic integers are pretty easy to define, but it's far from obvious what the point of them is, or how we should even think about them. In this video, I describe them as a practical tool: a collection of number systems that are related to the usual integers Z, but where solving equatio
From playlist [ANT] An unorthodox introduction to algebraic number theory
On a conjecture of Poonen and Voloch I: Probabilistic models(...) - Sawin - Workshop 1 - CEB T2 2019
Will Sawin (Columbia University) / 21.05.2019 On a conjecture of Poonen and Voloch I: Probabilistic models for counting rational points on random Fano hypersurfaces Poonen and Voloch have conjectured that almost every degree d Fano hypersur- face in Pn defined over the field of rational
From playlist 2019 - T2 - Reinventing rational points
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