Theorems in abstract algebra | Field (mathematics)
In field theory, the primitive element theorem is a result characterizing the finite degree field extensions that can be generated by a single element. Such a generating element is called a primitive element of the field extension, and the extension is called a simple extension in this case. The theorem states that a finite extension is simple if and only if there are only finitely many intermediate fields. An older result, also often called "primitive element theorem", states that every finite separable extension is simple; it can be seen as a consequence of the former theorem. These theorems imply in particular that all algebraic number fields over the rational numbers, and all extensions in which both fields are finite, are simple. (Wikipedia).
Using the property of equality of powers to solve an equation with exponents
👉 Learn how to solve exponential equations. An exponential equation is an equation in which a variable occurs as an exponent. To solve an exponential equation, we isolate the exponential part of the equation. Then we take the log of both sides. Note that the base of the log should correspo
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How to use the same base to solve an equation with exponents
👉 Learn how to solve exponential equations. An exponential equation is an equation in which a variable occurs as an exponent. To solve an exponential equation, we isolate the exponential part of the equation. Then we take the log of both sides. Note that the base of the log should correspo
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Solving a two step equation with a rational expressions
👉 Learn how to solve two step rational linear equations. A linear equation is an equation whose highest exponent on its variable(s) is 1. A rational equation is an equation containing at least one fraction whose numerator and (or) denominator are polynomials. To solve for a variable in a
From playlist Solve Two Step Equations with a Rational Fraction
Solving two step equations with a rational expression on one side
👉 Learn how to solve two step rational linear equations. A linear equation is an equation whose highest exponent on its variable(s) is 1. A rational equation is an equation containing at least one fraction whose numerator and (or) denominator are polynomials. To solve for a variable in a
From playlist Solve Two Step Equations with a Rational Fraction
Using one to one property with different bases to solve an exponential equation
👉 Learn how to solve exponential equations. An exponential equation is an equation in which a variable occurs as an exponent. To solve an exponential equation, we isolate the exponential part of the equation. Then we take the log of both sides. Note that the base of the log should correspo
From playlist Solve Exponential Equations without a Calculator
Solving for x when your variable is on the right side and being divided
👉 Learn how to solve two step rational linear equations. A linear equation is an equation whose highest exponent on its variable(s) is 1. A rational equation is an equation containing at least one fraction whose numerator and (or) denominator are polynomials. To solve for a variable in a
From playlist Solve Two Step Equations with a Rational Fraction
Solving a two step equation rational equation
👉 Learn how to solve two step rational linear equations. A linear equation is an equation whose highest exponent on its variable(s) is 1. A rational equation is an equation containing at least one fraction whose numerator and (or) denominator are polynomials. To solve for a variable in a
From playlist Solve Two Step Equations with a Rational Fraction
Using the property of equality to solve equations with exponents
👉 Learn how to solve exponential equations. An exponential equation is an equation in which a variable occurs as an exponent. To solve an exponential equation, we isolate the exponential part of the equation. Then we take the log of both sides. Note that the base of the log should correspo
From playlist Solve Exponential Equations without a Calculator
This lecture is part of an online undergraduate course on the theory of numbers. We prove Wilsons' theorem that (p-1)! = -1 mod p, and give some generalizations and applications of it. For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj52Qf7lc3H
From playlist Theory of numbers
Theory of numbers: Congruences: Primitive roots
This lecture is part of an online undergraduate course on the theory of numbers. We define primitive roots, and show that all primes have primitive roots. Correction: At 16:00 two of the square roots of 1 should be (2^k)/2 +1, (2^k)/2-1, not 2^k/2, -2^k/2. For the other lectures in th
From playlist Theory of numbers
Introduction to number theory lecture 23. Primitive roots.
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 show that every prime has a primitive root. The textbook is "An introduction to the the
From playlist Introduction to number theory (Berkeley Math 115)
Lecture 34. Galois groups and fixed fields
From playlist Abstract Algebra 2
Introduction to number theory lecture 30. Fields in number theory
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 extend some of the results we proved about the integers mod p to more general fields.
From playlist Introduction to number theory (Berkeley Math 115)
Solving a two step equation with division
👉 Learn how to solve two step rational linear equations. A linear equation is an equation whose highest exponent on its variable(s) is 1. A rational equation is an equation containing at least one fraction whose numerator and (or) denominator are polynomials. To solve for a variable in a
From playlist Solve Two Step Equations with a Rational Fraction
Foundations - Seminar 11 - Gödel's incompleteness theorem Part 3
Billy Price and Will Troiani present a series of seminars on foundations of mathematics. In this seminar Will Troiani continues with the proof of Gödel's incompleteness theorem, discussing Gödel's beta function and the role of the Chinese Remainder theorem in the incompleteness theorem. Y
From playlist Foundations seminar
Number Theory: Primitive Roots - Oxford Mathematics 2nd Year Student Lecture
Like many Universities around the world, Oxford has gone online for lockdown. So how do our student lectures look? In this, the second online lecture we are making widely available, world-renowned mathematician Ben Green introduces and delivers a short lecture on Primitive Roots, part of t
From playlist Oxford Mathematics 2nd Year Student Lectures
Lie groups: Poincare-Birkhoff-Witt theorem
This lecture is part of an online graduate course on Lie groups. We state the Poincare-Birkhoff Witt theorem, which shows that the universal enveloping algebra (UEA) of a Lie algebra is the same size as a polynomial algebra. We prove it for Lie algebras of Lie groups and sketch a proof of
From playlist Lie groups
Laurent Fargues - Courbes et fibrés vectoriels en théorie de Hodge p-adique
Courbes et fibrés vectoriels en théorie de Hodge p-adique
From playlist 28ème Journées Arithmétiques 2013
Peter Stevenhagen: Character sums for primitive root densities
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Jean-Morlet Chair - Shparlinski/Kohel
Learning to use the one to one property to solve an exponential equation
👉 Learn how to solve exponential equations. An exponential equation is an equation in which a variable occurs as an exponent. To solve an exponential equation, we isolate the exponential part of the equation. Then we take the log of both sides. Note that the base of the log should correspo
From playlist Solve Exponential Equations without a Calculator