Class field theory | Group theory
In the mathematical field of algebraic number theory, the concept of principalization refers to a situation when, given an extension of algebraic number fields, some ideal (or more generally fractional ideal) of the ring of integers of the smaller field isn't principal but its extension to the ring of integers of the larger field is. Its study has origins in the work of Ernst Kummer on ideal numbers from the 1840s, who in particular proved that for every algebraic number field there exists an extension number field such that all ideals of the ring of integers of the base field (which can always be generated by at most two elements) become principal when extended to the larger field. In 1897 David Hilbert conjectured that the maximal abelian unramified extension of the base field, which was later called the Hilbert class field of the given base field, is such an extension. This conjecture, now known as principal ideal theorem, was proved by Philipp Furtwängler in 1930 after it had been translated from number theory to group theory by Emil Artin in 1929, who made use of his general reciprocity law to establish the reformulation. Since this long desired proof was achieved by means of Artin transfers of non-abelian groups with derived length two, several investigators tried to exploit the theory of such groups further to obtain additional information on the principalization in intermediate fields between the base field and its Hilbert class field. The first contributions in this direction are due to Arnold Scholz and Olga Taussky in 1934, who coined the synonym capitulation for principalization. Another independent access to the principalization problem via Galois cohomology of unit groups is also due to Hilbert and goes back to the chapter on cyclic extensions of number fields of prime degree in his number report, which culminates in the famous Theorem 94. (Wikipedia).
Algebra for Beginners | Basics of Algebra
#Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. Table of Conten
From playlist Linear Algebra
Principal axes theorem + orthogonal matrices
Free ebook http://tinyurl.com/EngMathYT A basic introduction to orthogonal matrices and the principal axes theorem. Several examples are presented involving a simplification of quadratic (quadric) forms. A proof is also given.
From playlist Engineering Mathematics
Math 060 Linear Algebra 28 111914: Diagonalization of Matrices
Diagonalization of matrices; equivalence of diagonalizability with existence of an eigenvector basis; example of diagonalization; algebraic multiplicity; geometric multiplicity; relation between the two (geometric cannot exceed algebraic).
From playlist Course 4: Linear Algebra
Linear Algebra Full Course for Beginners to Experts
Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis may be basically viewed as the application of l
From playlist Linear Algebra
How do we add matrices. A matrix is an abstract object that exists in its own right, and in this sense, it is similar to a natural number, or a complex number, or even a polynomial. Each element in a matrix has an address by way of the row in which it is and the column in which it is. Y
From playlist Introducing linear algebra
In this tutorial we take a look at elementary matrices. They start life off as identity matrices to which a single elementary row operation is performed. They form the building blocks of Gauss-Jordan elimination. In a future video we will use the to do LU decomposition of matrices.
From playlist Introducing linear algebra
Linear Algebra for Beginners | Linear algebra for machine learning
Linear algebra is the branch of mathematics concerning linear equations such as linear functions and their representations through matrices and vector spaces. Linear algebra is central to almost all areas of mathematics. In this course you will learn most of the basics of linear algebra wh
From playlist Linear Algebra
Most math students don’t understand this about square roots!
How to find the square root of a number - also what is the principal square root. For more in-depth math help check out my catalog of courses. Every course includes over 275 videos of easy to follow and understand math instruction, with fully explained practice problems and printable work
From playlist GED Prep Videos
Principal Component Analysis (PCA) — Topic 40 of Machine Learning Foundations
Via highly visual hands-on code demos in Python, this video introduces Principal Component Analysis, a prevalent and powerful machine learning technique for finding patterns in unlabeled data. There are eight subjects covered comprehensively in the ML Foundations series and this video is
From playlist Linear Algebra for Machine Learning
Equivariant principal bundles on toric varieties- Part 1 by Mainak Poddar
DISCUSSION MEETING ANALYTIC AND ALGEBRAIC GEOMETRY DATE:19 March 2018 to 24 March 2018 VENUE:Madhava Lecture Hall, ICTS, Bangalore. Complex analytic geometry is a very broad area of mathematics straddling differential geometry, algebraic geometry and analysis. Much of the interactions be
From playlist Analytic and Algebraic Geometry-2018
Square Roots - 5 Things you MUST Know!
Square roots in math - 5 things every algebra student needs to understand. For more in-depth math help check out my catalog of courses. Every course includes over 275 videos of easy to follow and understand math instruction, with fully explained practice problems and printable worksheets
From playlist GED Prep Videos
Multiply and simplify a radical expression 2 | Algebra I | Khan Academy
Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/algebra-home/alg-exp-and-log/miscellaneous-radicals/v/multiply-and-simplify-a-radical-expression-2 Multiply and Simply a Radical Expression Watch the next lesson
From playlist Exponent expressions and equations | Algebra I | Khan Academy
Minhyong Kim: Recent progress on the effective Mordell problem
SMRI Algebra and Geometry Online: Minhyong Kim (University of Warwick) Abstract: In 1983, Gerd Faltings proved the Mordell conjecture stating that curves of genus at least two have only finitely many rational points. This can be understood as the statement that most polynomial equations
From playlist SMRI Algebra and Geometry Online
Simplifying square roots | Exponent expressions and equations | Algebra I | Khan Academy
Simplifying Square Roots Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/algebra/exponent-equations/simplifying-radical-expressions/e/simplifying_radicals?utm_source=YT&utm_medium=Desc&utm_campaign=AlgebraI Watch the next lesson: https://www.k
From playlist Exponent expressions and equations | Algebra I | Khan Academy
9A_4 The Inverse of a Matrix Using the Determinant
Calculating the inverse of a matrix by use of the determinant of the matrix
From playlist Linear Algebra
Extraneous solutions to radical equations | Algebra I | Khan Academy
Extraneous Solutions to Radical Equations Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/algebra/exponent-equations/radical_equations/e/radical_equations?utm_source=YT&utm_medium=Desc&utm_campaign=AlgebraI Watch the next lesson: https://www.k
From playlist Algebra II | High School Math | Khan Academy