In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraic number field. More specifically, it is proportional to the squared volume of the fundamental domain of the ring of integers, and it regulates which primes are ramified. The discriminant is one of the most basic invariants of a number field, and occurs in several important analytic formulas such as the functional equation of the Dedekind zeta function of K, and the analytic class number formula for K. A theorem of Hermite states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an open problem, and the subject of current research. The discriminant of K can be referred to as the absolute discriminant of K to distinguish it from the relative discriminant of an extension K/L of number fields. The latter is an ideal in the ring of integers of L, and like the absolute discriminant it indicates which primes are ramified in K/L. It is a generalization of the absolute discriminant allowing for L to be bigger than Q; in fact, when L = Q, the relative discriminant of K/Q is the principal ideal of Z generated by the absolute discriminant of K. (Wikipedia).
This lecture is part of an online graduate course on Galois theory. We define the discriminant of a finite field extension, ans show that it is essentially the same as the discriminant of a minimal polynomial of a generator. We then give some applications to algebraic number fields. Corr
From playlist Galois theory
What is the discriminant and what does it mean
👉 Learn all about the discriminant of quadratic equations. A quadratic equation is an equation whose highest power on its variable(s) is 2. The discriminant of a quadratic equation is a formula which is used to determine the type of roots (solutions) the quadratic equation have. The disc
From playlist Discriminant of a Quadratic Equation | Learn About
How to use the discriminat to describe your solutions
👉 Learn how to determine the discriminant of quadratic equations. A quadratic equation is an equation whose highest power on its variable(s) is 2. The discriminant of a quadratic equation is a formula which is used to determine the type of roots (solutions) the quadratic equation have. T
From playlist Discriminant of a Quadratic Equation
Determine and describe the discriminant
👉 Learn how to determine the discriminant of quadratic equations. A quadratic equation is an equation whose highest power on its variable(s) is 2. The discriminant of a quadratic equation is a formula which is used to determine the type of roots (solutions) the quadratic equation have. T
From playlist Discriminant of a Quadratic Equation
What is the formula for a perfect square trinomial and how does the discriminant fit in
👉 Learn all about the discriminant of quadratic equations. A quadratic equation is an equation whose highest power on its variable(s) is 2. The discriminant of a quadratic equation is a formula which is used to determine the type of roots (solutions) the quadratic equation have. The disc
From playlist Discriminant of a Quadratic Equation | Learn About
What does the discriminat tell us about our zeros graphically
👉 Learn all about the discriminant of quadratic equations. A quadratic equation is an equation whose highest power on its variable(s) is 2. The discriminant of a quadratic equation is a formula which is used to determine the type of roots (solutions) the quadratic equation have. The disc
From playlist Discriminant of a Quadratic Equation | Learn About
What does the discriminant tell us about the solutions of a quadratic equation
👉 Learn all about the discriminant of quadratic equations. A quadratic equation is an equation whose highest power on its variable(s) is 2. The discriminant of a quadratic equation is a formula which is used to determine the type of roots (solutions) the quadratic equation have. The disc
From playlist Discriminant of a Quadratic Equation | Learn About
How to find the discriminant of a quadratic and label the solutions
👉 Learn how to determine the discriminant of quadratic equations. A quadratic equation is an equation whose highest power on its variable(s) is 2. The discriminant of a quadratic equation is a formula which is used to determine the type of roots (solutions) the quadratic equation have. T
From playlist Discriminant of a Quadratic Equation
Overview of solutions of a quadratic function and the discriminant
👉 Learn all about the discriminant of quadratic equations. A quadratic equation is an equation whose highest power on its variable(s) is 2. The discriminant of a quadratic equation is a formula which is used to determine the type of roots (solutions) the quadratic equation have. The disc
From playlist Discriminant of a Quadratic Equation | Learn About
Tobias Braun - Orthogonal Determinants
Basic concepts and notions of orthogonal representations are in- troduced. If X : G → GL(V ) is a K-representation of a nite group G it may happen that its image X(G) xes a non-degenerate quadratic form q on V . In this case X and its character χ : G → K, g 7 → trace(X(g)) are called ortho
From playlist École d'Été 2022 - Cohomology Geometry and Explicit Number Theory
CTNT 2018 - "The Tsfasman-Vladut Generalization of the Brauer-Siegel Theorem" by Farshid Hajir
This is lecture on "The Tsfasman-Vladut Generalization of the Brauer-Siegel Theorem", by Farshid Hajir (UMass Amherst), during CTNT 2018, the Connecticut Summer School in Number Theory. For more information about CTNT and other resources and notes, see https://ctnt-summer.math.uconn.edu/
From playlist CTNT 2018 - Guest Lectures
[ANT13] Dedekind domains, integral closure, discriminants... and some other loose ends
In this video, we see an example of how badly this theory can fail in a non-Dedekind domain, and so - regrettably - we finally break our vow of not learning what a Dedekind domain is.
From playlist [ANT] An unorthodox introduction to algebraic number theory
Asymptotics of number fields - Manjul Bhargava [2011]
Asymptotics of number fields Introductory Workshop: Arithmetic Statistics January 31, 2011 - February 04, 2011 January 31, 2011 (11:40 AM PST - 12:40 PM PST) Speaker(s): Manjul Bhargava (Princeton University) Location: MSRI: Simons Auditorium http://www.msri.org/workshops/566
From playlist Number Theory
Xevi Guitart : Endomorphism algebras of geometrically split abelian surfaces over Q
CONFERENCE Recording during the thematic meeting : "COUNT, COmputations and their Uses in Number Theory" the February 28, 2023 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Jean Petit Find this video and other talks given by worldwide mathematici
From playlist JEAN MORLET CHAIR
Ari Shnidman: Monogenic cubic fields and local obstructions
Recording during the meeting "Zeta Functions" the December 05, 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 Mathematics Library: http:
From playlist Number Theory
Introduction to elliptic curves and BSD Conjecture by Sujatha Ramadorai
12 December 2016 to 22 December 2016 VENUE Madhava Lecture Hall, ICTS Bangalore The Birch and Swinnerton-Dyer conjecture is a striking example of conjectures in number theory, specifically in arithmetic geometry, that has abundant numerical evidence but not a complete general solution. An
From playlist Theoretical and Computational Aspects of the Birch and Swinnerton-Dyer Conjecture
Topological and arithmetic intersection numbers attached to real quadratic cycles -Henri Darmon
Workshop on Motives, Galois Representations and Cohomology Around the Langlands Program Topic: Topological and arithmetic intersection numbers attached to real quadratic cycles Speaker: Henri Darmon Affiliation: McGill University Date: November 8, 2017 For more videos, please visit http
From playlist Mathematics
How to identify the number of solutions using the discriminant of a quadratic
👉 Learn all about the discriminant of quadratic equations. A quadratic equation is an equation whose highest power on its variable(s) is 2. The discriminant of a quadratic equation is a formula which is used to determine the type of roots (solutions) the quadratic equation have. The disc
From playlist Discriminant of a Quadratic Equation | Learn About
CTNT 2020 - A virtual tour of Magma
This video is part of a series of videos on "Computations in Number Theory Research" that are offered as a mini-course during CTNT 2020. In this video, we take a virtual tour of Magma, the computational algebra system, paying special attention to its number theory capabilities. Please clic
From playlist CTNT 2020 - Computations in Number Theory Research