In field theory, a branch of algebra, an algebraic field extension is called a separable extension if for every , the minimal polynomial of over F is a separable polynomial (i.e., its formal derivative is not the zero polynomial, or equivalently it has no repeated roots in any extension field). There is also a more general definition that applies when E is not necessarily algebraic over F. An extension that is not separable is said to be inseparable. Every algebraic extension of a field of characteristic zero is separable, and every algebraic extension of a finite field is separable.It follows that most extensions that are considered in mathematics are separable. Nevertheless, the concept of separability is important, as the existence of inseparable extensions is the main obstacle for extending many theorems proved in characteristic zero to non-zero characteristic. For example, the fundamental theorem of Galois theory is a theorem about normal extensions, which remains true in non-zero characteristic only if the extensions are also assumed to be separable. The opposite concept, a purely inseparable extension, also occurs naturally, as every algebraic extension may be decomposed uniquely as a purely inseparable extension of a separable extension. An algebraic extension of fields of non-zero characteristics p is a purely inseparable extension if and only if for every , the minimal polynomial of over F is not a separable polynomial, or, equivalently, for every element x of E, there is a positive integer k such that . The simplest example of a (purely) inseparable extension is , fields of rational functions in the indeterminate x with coefficients in the finite field . The element has minimal polynomial , having and a p-fold multiple root, as . This is a simple algebraic extension of degree p, as , but it is not a normal extension since the Galois group is trivial. (Wikipedia).
Galois theory: Separable extensions
This lecture is part of an online graduate course on Galois theory. We define separable algebraic extensions, and give some examples of separable and non-separable extensions. At the end we briefly discuss purely inseparable extensions.
From playlist Galois theory
Applications with Separable Equations (Differential Equations 14)
https://www.patreon.com/ProfessorLeonard Using Separable Differential Equations to solve application problems involving Exponential Growth and Decay.
From playlist Differential Equations
Separable differential equations
Download the free PDF http://tinyurl.com/EngMathYT A basic lesson on how to solve separable differential equations. Such equations have important applications in the modelling of dynamic phenomena.
From playlist A second course in university calculus.
Separable Differential Equations (Differential Equations 12)
https://www.patreon.com/ProfessorLeonard How to solve Separable Differential Equations by Separation of Variables. Lots of examples!!
From playlist Differential Equations
Separable differential equation
Free ebook http://tinyurl.com/EngMathYT An example of how to solve separable differential equations.
From playlist A second course in university calculus.
More Solving Separable Differential Equations Calculus 1 AB
After reviewing the definition of a Separable Differential Equation, I work through 3 examples of finding general and particular solutions. EXAMPLES at 0:55 5:59 17:12 I would like to send out a HUGE THANK YOU to my YouTube student Jigyasa Nigam that did the Closed Captioning of this les
From playlist Calculus
Separable Differential Equation (example 1)
Learn how to solve a separable differential equation. This is usually the first kind of differential equations that we learn in an ordinary differential equations class. To learn how to solve different types of differential equations: Check out the differential equation playlist: 👉 https
From playlist Differential Equations: Separable (Nagle Sect2.2)
separable differential equation with an initial condition
Learn how to solve a separable differential equation with an initial condition. This is usually the first kind of differential equations that we learn in an ordinary differential equations class. To learn how to solve different types of differential equations: Check out the differential e
From playlist Differential Equations: Separable (Nagle Sect2.2)
Separable Differential Equations
Free ebook http://tinyurl.com/EngMathYT A lecture that introduces differential equations and show how to solve separable differential equations. Plenty of examples are discussed and solved. The ideas are useful in modelling phenomena.
From playlist A second course in university calculus.
CTNT 2022 - Local Fields (Lecture 4) - by Christelle Vincent
This video is part of a mini-course on "Local Fields" 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 - Local Fields (by Christelle Vincent)
Pushing back the barrier of imperfection - F-V. Kuhlmann - Workshop 2 - CEB T1 2018
Franz-Viktor Kuhlmann (Szczecin) / 06.03.2018 The word “imperfection” in our title not only refers to fields that are not perfect, but also to the defect of valued field extensions. The latter is not necessarily directly connected with imperfect fields but may always appear when at least
From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields
Kazuo Murota: Discrete Convex Analysis (Part 3)
The lecture was held within the framework of the Hausdorff Trimester Program: Combinatorial Optimization
From playlist HIM Lectures 2015
Galois theory: Galois extensions
This lecture is part of an online graduate course on Galois theory. We define Galois extensions in 5 different ways, and show that 4 f these conditions are equivalent. (The 5th equivalence will be proved in a later lecture.) We use this to show that any finite group is the Galois group of
From playlist Galois theory
Galois theory: Primitive elements
This lecture is part of an online graduate course on Galois theory. We show that any finite separable extension of fields has a primitive element (or generator) and given n example of a finite non-separable extension with no primitive elements.
From playlist Galois theory
FIT3.1.1. Roots of Polynomials
Field Theory: We recall basic factoring results for polynomials from Ring Theory and give a definition of a splitting field. This allows one to consider any irreducible polynomial as a set of roots, and in turn we consider when an irreducible polynomial can have multiple roots. We finish
From playlist Abstract Algebra
Galois theory: Infinite Galois extensions
This lecture is part of an online graduate course on Galois theory. We show how to extend Galois theory to infinite Galois extensions. The main difference is that the Galois group has a topology, and intermediate field extensions now correspond to closed subgroups of the Galois group. We
From playlist Galois theory
James Oxley: A matroid extension result
Abstract: Let (A,B) be a 3-separation in a matroid M. If M is representable, then, in the underlying projective space, there is a line where the subspaces spanned by A and B meet, and M can be extended by adding elements from this line. In general, Geelen, Gerards, and Whittle proved that
From playlist Combinatorics
Title: Strongly Étale Difference Algebras and Babbitt's Decomposition
From playlist Fall 2015
1_4 Exponential Growth and Decay
Examples of first order separable differential equations
From playlist Advanced Calculus / Multivariable Calculus