Algebraic number field

What is a field ?

Definition of a Field In this video, I define the concept of a field, which is basically any set where you can add, subtract, add, and divide things. Then I show some neat properties that have to be true in fields. Enjoy! What is an Ordered Field: https://youtu.be/6mc5E6x7FMQ Check out

From playlist Real Numbers

FIT2.3.1. Algebraic Numbers

Field Theory: We consider the property of algebraic in terms of finite degree, and we define algebraic numbers as those complex numbers that are algebraic over the rationals. Then we give an overview of algebraic numbers with examples.

From playlist Abstract Algebra

Number theory Full Course [A to Z]

Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure #mathematics devoted primarily to the study of the integers and integer-valued functions. Number theorists study prime numbers as well as the properties of objects made out of integers (for example, ratio

From playlist Number Theory

Field Theory - Algebraically Closed Fields - Lecture 9

In this video we define what an algebraically closed field and assert without proof that they exist. We also explain why if you can find a single root for any polynomial, then you can find them all.

From playlist Field Theory

Algebraic number theory and rings I | Math History | NJ Wildberger

In the 19th century, algebraists started to look at extension fields of the rational numbers as new domains for doing arithmetic. In this way the notion of an abstract ring was born, through the more concrete examples of rings of algebraic integers in number fields. Key examples include

From playlist MathHistory: A course in the History of Mathematics

Field Theory: Definition/ Axioms

This video is about the basics axioms of fields.

From playlist Basics: Field Theory

Algebraic number theory and rings II | Math History | NJ Wildberger

In the 19th century, algebraists started to look at extension fields of the rational numbers as new domains for doing arithmetic. In this way the notion of an abstract ring was born, through the more concrete examples of rings of algebraic integers in number fields. Key examples include

From playlist MathHistory: A course in the History of Mathematics

FIT1.1. Number Fields

Field Theory: We give a brief review of some of the main results on fields in basic ring theory and give examples to motivate field theory. Examples include field automorphisms for the rational polynomials x^2-2 and x^3-2.

From playlist Abstract Algebra

The Structure of Fields: What is a field and a connection between groups and fields

This video is primarily meant to help develop some ideas around the structure of fields and a connection between groups and fields (which will allow me to create more abstract algebra videos in the future! 😀😅🤓) 00:00 Intro 01:04 What is a Field? Here we give the definition of a field in

From playlist The New CHALKboard

Galois theory: Algebraic closure

This lecture is part of an online graduate course on Galois theory. We define the algebraic closure of a field as a sort of splitting field of all polynomials, and check that it is algebraically closed. We hen give a topological proof that the field C of complex numbers is algebraically

From playlist Galois theory

algebraic geometry 30 The Ax Grothendieck theorem

This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the Ax-Grothendieck theorem, which states that an injective regular map between varieties is surjective. The proof uses a strange technique: first prove the resu

From playlist Algebraic geometry I: Varieties

Galois theory: Transcendental extensions

This lecture is part of an online graduate course on Galois theory. We describe transcendental extension of fields and transcendence bases. As applications we classify algebraically closed fields and show hw to define the dimension of an algebraic variety.

From playlist Galois theory

Lie Groups and Lie Algebras: Lesson 2 - Quaternions

This video is about Lie Groups and Lie Algebras: Lesson 2 - Quaternions We study the algebraic nature of quaternions and cover the ideas of an algebra and a field. Later we will discover how quaternions fit into the description of the classical Lie Groups. NOTE: An astute viewer noted th

From playlist Lie Groups and Lie Algebras

On the notion of genus for division algebras and algebraic groups - Andrei Rapinchu

Joint IAS/Princeton University Number Theory Seminar Topic: On the notion of genus for division algebras and algebraic groups Speaker: Andrei Rapinchu Affiliation: University of Virginia Date: November 2, 2017 For more videos, please visit http://video.ias.edu

From playlist Mathematics

Galois theory: Field extensions

This lecture is part of an online course on Galois theory. We review some basic results about field extensions and algebraic numbers. We define the degree of a field extension and show that a number is algebraic over a field if and only if it is contained in a finite extension. We use thi

From playlist Galois theory

FIT2.3.3. Algebraic Extensions

Field Theory: We define an algebraic extension of a field F and show that successive algebraic extensions are also algebraic. This gives a useful criterion for checking algberaic elements. We finish with algebraic closures.

From playlist Abstract Algebra

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

A geometric model for the bounded derived category of a gentle algebra, Sibylle Schroll Lecture 3

Gentle algebras are quadratic monomial algebras whose representation theory is well understood. In recent years they have played a central role in several different subjects such as in cluster algebras where they occur as Jacobian algebras of quivers with potentials obtained from triangula

From playlist Winter School on “Connections between representation Winter School on “Connections between representation theory and geometry"

Real Numbers

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From playlist Integers

SummerSchool 20060724 1130 Starr - Tsen-Lang theorem

From playlist Clay Mathematics Institute Summer School 2006 on "Arithmetic geometry"