Theorems in abstract algebra | Field (mathematics)

Primitive element theorem

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).

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Using the property of equality of powers to solve an equation with exponents

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How to use the same base to solve an equation with exponents

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Solving two step equations with a rational expression on one side

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From playlist Solve Two Step Equations with a Rational Fraction

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Using one to one property with different bases to solve an exponential equation

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Solving for x when your variable is on the right side and being divided

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From playlist Solve Two Step Equations with a Rational Fraction

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Solving a two step equation rational equation

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From playlist Solve Two Step Equations with a Rational Fraction

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Using the property of equality to solve equations with exponents

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From playlist Solve Exponential Equations without a Calculator

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Wilson's theorem

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

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Theory of numbers: Congruences: Primitive roots

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From playlist Theory of numbers

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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)

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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)

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Solving a two step equation with division

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Foundations - Seminar 11 - Gödel's incompleteness theorem Part 3

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Number Theory: Primitive Roots - Oxford Mathematics 2nd Year Student Lecture

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From playlist Oxford Mathematics 2nd Year Student Lectures

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Lie groups: Poincare-Birkhoff-Witt theorem

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From playlist Lie groups

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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

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Learning to use the one to one property to solve an exponential equation

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Galois theory | Multiplicative group | Rational function | Vector space | Finite field | Frobenius endomorphism | Algebraic number field | Emil Artin | Rational number | Separable extension | Siméon Denis Poisson | Splitting field | Field extension | System of linear equations | Field theory (mathematics) | Characteristic (algebra) | Fundamental theorem of Galois theory | Pigeonhole principle | Cyclic group | Primitive element (finite field) | Simple extension | Linear combination | Basis (linear algebra) | Galois group | Joseph-Louis Lagrange | Ernst Steinitz | Degree of a field extension