Theorems in algebraic number theory
In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring OK of algebraic integers of a number field K. The regulator is a positive real number that determines how "dense" the units are. The statement is that the group of units is finitely generated and has rank (maximal number of multiplicatively independent elements) equal to r = r1 + r2 − 1 where r1 is the number of real embeddings and r2 the number of conjugate pairs of complex embeddings of K. This characterisation of r1 and r2 is based on the idea that there will be as many ways to embed K in the complex number field as the degree ; these will either be into the real numbers, or pairs of embeddings related by complex conjugation, so that n = r1 + 2r2. Note that if K is Galois over then either r1 = 0 or r2 = 0. Other ways of determining r1 and r2 are * use the primitive element theorem to write , and then r1 is the number of conjugates of α that are real, 2r2 the number that are complex; in other words, if f is the minimal polynomial of α over , then r1 is the number of real roots and 2r2 is the number of non-real complex roots of f (which come in complex conjugate pairs); * write the tensor product of fields as a product of fields, there being r1 copies of and r2 copies of . As an example, if K is a quadratic field, the rank is 1 if it is a real quadratic field, and 0 if an imaginary quadratic field. The theory for real quadratic fields is essentially the theory of Pell's equation. The rank is positive for all number fields besides and imaginary quadratic fields, which have rank 0. The 'size' of the units is measured in general by a determinant called the regulator. In principle a basis for the units can be effectively computed; in practice the calculations are quite involved when n is large. The torsion in the group of units is the set of all roots of unity of K, which form a finite cyclic group. For a number field with at least one real embedding the torsion must therefore be only {1,−1}. There are number fields, for example most imaginary quadratic fields, having no real embeddings which also have {1,−1} for the torsion of its unit group. Totally real fields are special with respect to units. If L/K is a finite extension of number fields with degree greater than 1 andthe units groups for the integers of L and K have the same rank then K is totally real and L is a totally complex quadratic extension. The converse holds too. (An example is K equal to the rationals and L equal to an imaginary quadratic field; both have unit rank 0.) The theorem not only applies to the maximal order OK but to any order O ⊂ OK. There is a generalisation of the unit theorem by Helmut Hasse (and later Claude Chevalley) to describe the structure of the group of S-units, determining the rank of the unit group in localizations of rings of integers. Also, the Galois module structure of has been determined. (Wikipedia).
What is the formula for the unit vector
http://www.freemathvideos.com In this video series I will show you how to find the unit vector when given a vector in component form and as a linear combination. A unit vector is simply a vector with the same direction but with a magnitude of 1 and an initial point at the origin. It is i
From playlist Vectors
Given a Vector in Component Form, Find the Unit Vector
Learn how to determine the unit vector of a vector in the same direction. The unit vector is a vector that has a magnitude of 1. The unit vector is obtained by dividing the given vector by its magnitude. #trigonometry#vectors #vectors
From playlist Vectors
Finding the Unit Vector of a Vector in Standard Form
Learn how to determine the unit vector of a vector in the same direction. The unit vector is a vector that has a magnitude of 1. The unit vector is obtained by dividing the given vector by its magnitude. #trigonometry#vectors #vectors
From playlist Vectors
Learn How to Determine the Unit Vector with the Same Direction as Another Vector
Learn how to determine the unit vector of a vector in the same direction. The unit vector is a vector that has a magnitude of 1. The unit vector is obtained by dividing the given vector by its magnitude. #trigonometry#vectors #vectors
From playlist Vectors
What is the formula for a unit vector from a vector in component form
http://www.freemathvideos.com In this video series I will show you how to find the unit vector when given a vector in component form and as a linear combination. A unit vector is simply a vector with the same direction but with a magnitude of 1 and an initial point at the origin. It is i
From playlist Vectors
How to find the point on the unit circle from the given real number
👉 Learn how to find the point on the unit circle given the angle of the point. A unit circle is a circle whose radius is 1. Given an angle in radians, to find the coordinate of points on the unit circle made by the given angle with the x-axis at the center of the unit circle, we plot the a
From playlist Evaluate Trigonometric Functions With The Unit Circle (ALG2)
Multivariable Calculus | Unit Vectors
We define a unit vector, the unit basis vectors, and give some associated examples. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Vectors for Multivariable Calculus
CTNT 2022 - 100 Years of Chebotarev Density (Lecture 1) - by Keith Conrad
This video is part of a mini-course on "100 Years of Chebotarev Density" that was taught during CTNT 2022, the Connecticut Summer School and Conference in Number Theory. More about CTNT: https://ctnt-summer.math.uconn.edu/
From playlist CTNT 2022 - 100 Years of Chebotarev Density (by Keith Conrad)
Low moments of character sums - Adam Harper
Joint IAS/Princeton University Number Theory Seminar Topic: Low moments of character sums Speaker: Adam Harper Affiliation: University of Warwick Date: April 08, 2021 For more video please visit http://video.ias.edu
From playlist Mathematics
Learning to determine the point on the unit circle by sketching the angle
👉 Learn how to find the point on the unit circle given the angle of the point. A unit circle is a circle whose radius is 1. Given an angle in radians, to find the coordinate of points on the unit circle made by the given angle with the x-axis at the center of the unit circle, we plot the a
From playlist Evaluate Trigonometric Functions With The Unit Circle (ALG2)
Adam Skalski: Translation invariant noncommutative Dirichlet forms
Talk by Adam Skalski in Global Noncommutative Geometry Seminar (Europe) http://www.noncommutativegeometry.nl/ncgseminar/ on April 28, 2021
From playlist Global Noncommutative Geometry Seminar (Europe)
Why do prime numbers make these spirals? | Dirichlet’s theorem, pi approximations, and more
A curious pattern, approximations for pi, and prime distributions. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: http://3b1b.co/spiral-thanks Based on this Math
From playlist Neat proofs/perspectives
Nikolay Moshchevitin: Diophantine exponents, best approximation and badly approximable numbers
CIRM VIRTUAL CONFERENCE Recorded during the meeting " Diophantine Problems, Determinism and Randomness" the November 23, 2020 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide
From playlist Virtual Conference
Lecture 23: The Dirichlet Problem on an Interval
MIT 18.102 Introduction to Functional Analysis, Spring 2021 Instructor: Dr. Casey Rodriguez View the complete course: https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/ YouTube Playlist: https://www.youtube.com/watch?v=QTg7040uSc0&list=PLUl4u3cNGP63micsJp_
From playlist MIT 18.102 Introduction to Functional Analysis, Spring 2021
CTNT 2022 - 100 Years of Chebotarev Density (Lecture 3) - by Keith Conrad
This video is part of a mini-course on "100 Years of Chebotarev Density" that was taught during CTNT 2022, the Connecticut Summer School and Conference in Number Theory. More about CTNT: https://ctnt-summer.math.uconn.edu/
From playlist CTNT 2022 - 100 Years of Chebotarev Density (by Keith Conrad)
Learn how to find the point of the unit circle when given a specific angle
👉 Learn how to find the point on the unit circle given the angle of the point. A unit circle is a circle whose radius is 1. Given an angle in radians, to find the coordinate of points on the unit circle made by the given angle with the x-axis at the center of the unit circle, we plot the a
From playlist Evaluate Trigonometric Functions With The Unit Circle (ALG2)
Introduction to number theory lecture 50. Dirichlet characters
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 review some properties of Dirichlet characters in preparation for the proof of Dirichlet
From playlist Introduction to number theory (Berkeley Math 115)
CTNT 2022 - 100 Years of Chebotarev Density (Lecture 2) - by Keith Conrad
This video is part of a mini-course on "100 Years of Chebotarev Density" that was taught during CTNT 2022, the Connecticut Summer School and Conference in Number Theory. More about CTNT: https://ctnt-summer.math.uconn.edu/
From playlist CTNT 2022 - 100 Years of Chebotarev Density (by Keith Conrad)
Andrew Wiles, Twenty years of the Birch--Swinnerton-Dyer conjecture
2018 Clay Research Conference, CMI at 20
From playlist CMI at 20
Given a vector find the unit vector u and check your answer
Learn how to determine the unit vector of a vector in the same direction. The unit vector is a vector that has a magnitude of 1. The unit vector is obtained by dividing the given vector by its magnitude. #trigonometry#vectors #vectors
From playlist Vectors