Galois theory: Splitting fields
This lecture is part of an online course on Galois theory. We define the splitting field of a polynomial p over a field K (a field that is generated by roots of p and such that p splits into linear factors). We give a few examples, and show that it exists and is unique up to isomorphism
From playlist Galois theory
FIT4.3.2. Example of Galois Group over Finite Field
Field Theory: We compare the splitting fields of the polynomial f(x)=x^8-1 over the rationals and Z/5. We compute the Galois groups and identify Galois correspondences.
From playlist Abstract Algebra
FIT4.3.1. Galois Group of Order 8
Field Theory: Let K be Q[sqrt(2), sqrt(3), sqrt(5)], the splitting field of f(x) = (x^2-2)(x^2-3)(x^2-5) over Q. Find the Galois group of K over Q, find all subgroups of the Galois group, and find all subfields of K over Q.
From playlist Abstract Algebra
Galois theory: Normal extensions
This lecture is part of an online graduate course on Galois theory. We define normal extensions of fields by three equivalent conditions, and give some examples of normal and non-normal extensions. In particular we show that a normal extension of a normal extension need not be normal.
From playlist Galois theory
Field Theory - the Extension Lemma - Lecture 14
--Let \sigma: F_1 \to F_2 be an isomorphism of fields. --Let f_1(x) \in F_1[x] and let f_2(x) be the image in f_1(x) under the natural isomorphism F_1[x] \to F_2[x]. --Let L_1 is the splitting field of f_1(x) and L_2 the splitting field of f_2(x). Lemma: The isomorphism \sigma extends t
From playlist Field Theory
This lecture is part of an online graduate course on Galois theory. We use the theory of splitting fields to classify finite fields: there is one of each prime power order (up to isomorphism). We give a few examples of small order, and point out that there seems to be no good choice for
From playlist Galois theory
Galois theory: Infinite Galois extensions
This lecture is part of an online graduate course on Galois theory. We show how to extend Galois theory to infinite Galois extensions. The main difference is that the Galois group has a topology, and intermediate field extensions now correspond to closed subgroups of the Galois group. We
From playlist Galois theory
CTNT 2020 - Infinite Galois Theory (by Keith Conrad) - Lecture 4
The Connecticut Summer School in Number Theory (CTNT) is a summer school in number theory for advanced undergraduate and beginning graduate students, to be followed by a research conference. For more information and resources please visit: https://ctnt-summer.math.uconn.edu/
From playlist CTNT 2020 - Infinite Galois Theory (by Keith Conrad)
Lecture 34. Galois groups and fixed fields
From playlist Abstract Algebra 2
Explicit formulae for Gross-Stark units and Hilbert’s 12th problem by Mahesh Kakde
PERFECTOID SPACES ORGANIZERS: Debargha Banerjee, Denis Benois, Chitrabhanu Chaudhuri, and Narasimha Kumar Cheraku DATE & TIME: 09 September 2019 to 20 September 2019 VENUE: Madhava Lecture Hall, ICTS, Bangalore Scientific committee: Jacques Tilouine (University of Paris, France) Eknath
From playlist Perfectoid Spaces 2019
FIT4.1. Galois Group of a Polynomial
EDIT: There was an in-video annotation that was erased in 2018. My source (Herstein) assumes characteristic 0 for the initial Galois theory section, so separability is an automatic property. Let's assume that unless noted. In general, Galois = separable plus normal. Field Theory: We
From playlist Abstract Algebra
Modular symbols and arithmetic - Romyar Sharifi
Locally Symmetric Spaces Seminar Topic: A Modular symbols and arithmetic Speaker: Romyar Sharifi Affiliation: niversity of California; Member, School of Mathematics Date: January 23, 2018 For more videos, please visit http://video.ias.edu
From playlist Mathematics
Emmy Noether: breathtaking mathematics - Georgia Benkart
Celebrating Emmy Noether Topic: Emmy Noether: breathtaking mathematics Speaker: Georgia Benkart Affiliation: University of Wisconsin-Madison Date: Friday, May 6 By the mid 1920s, Emmy Noether had made fundamental contributions to commutative algebra and to the theory of invariants.
From playlist Celebrating Emmy Noether
Iwasawa theory of the fine Selmer groups of Galois representations by Sujatha Ramdorai
PERFECTOID SPACES ORGANIZERS: Debargha Banerjee, Denis Benois, Chitrabhanu Chaudhuri, and Narasimha Kumar Cheraku DATE & TIME: 09 September 2019 to 20 September 2019 VENUE: Madhava Lecture Hall, ICTS, Bangalore Scientific committee: Jacques Tilouine (University of Paris, France) Eknath
From playlist Perfectoid Spaces 2019
Donald Cartwright : Construction of lattices defining fake projective planes - lecture 2
Recording during the meeting "Ball Quotient Surfaces and Lattices " the February 25, 2019 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Ma
From playlist Algebraic and Complex Geometry
Kevin Buzzard (lecture 11/20) Automorphic Forms And The Langlands Program [2017]
Full course playlist: https://www.youtube.com/playlist?list=PLhsb6tmzSpiysoRR0bZozub-MM0k3mdFR http://wwwf.imperial.ac.uk/~buzzard/MSRI/ Summer Graduate School Automorphic Forms and the Langlands Program July 24, 2017 - August 04, 2017 Kevin Buzzard (Imperial College, London) https://w
From playlist MSRI Summer School: Automorphic Forms And The Langlands Program, by Kevin Buzzard [2017]
FIT4.2. Automorphisms and Degree
EDIT: As noted before, my source (Herstein) assumes characteristic 0 for the initial Galois theory section. We'll assume separability unless noted. Note that Galois = separable plus normal. Field Theory: Having established an estimate on the size of a Galois group of a polynomial, w
From playlist Abstract Algebra
Kevin Buzzard (lecture 2/20) Automorphic Forms And The Langlands Program [2017]
Full course playlist: https://www.youtube.com/playlist?list=PLhsb6tmzSpiysoRR0bZozub-MM0k3mdFR http://wwwf.imperial.ac.uk/~buzzard/MSRI/ Summer Graduate School Automorphic Forms and the Langlands Program July 24, 2017 - August 04, 2017 Kevin Buzzard (Imperial College, London) https://w
From playlist MSRI Summer School: Automorphic Forms And The Langlands Program, by Kevin Buzzard [2017]
Winnie Li: Unramified graph covers of finite degree
Abstract: Given a finite connected undirected graph X, its fundamental group plays the role of the absolute Galois group of X. The familiar Galois theory holds in this setting. In this talk we shall discuss graph theoretical counter parts of several important theorems for number fields. T
From playlist Women at CIRM
FIT4.3. Galois Correspondence 1 - Examples
Field Theory: We define Galois extensions and state the Fundamental Theorem of Galois Theory. Proofs are given in the next part; we give examples to illustrate the main ideas.
From playlist Abstract Algebra