Field Theory: Let F be a subfield of the field K. We consider K as a vector space over F and define the degree of K over F as the dimension. We give a degree formula for successive extensions, and consider extensions in terms of bases. EDIT: Typo - around 3:15, it should be cube root(2
From playlist Abstract Algebra
This video is about extensions of fields.
From playlist Basics: Field Theory
FIT2.3.3. Algebraic Extensions
Field Theory: We define an algebraic extension of a field F and show that successive algebraic extensions are also algebraic. This gives a useful criterion for checking algberaic elements. We finish with algebraic closures.
From playlist Abstract Algebra
Galois theory: Field extensions
This lecture is part of an online course on Galois theory. We review some basic results about field extensions and algebraic numbers. We define the degree of a field extension and show that a number is algebraic over a field if and only if it is contained in a finite extension. We use thi
From playlist Galois theory
Field Theory: We consider the case of simple extensions, where we adjoin a single element to a given field. The cases of transcendental and algebraic arise, depending on whether the kernel of the evaluation map is zero or not. In the algebraic case, we define the minimal polynomial, show
From playlist Abstract Algebra
Degrees in Towers - Field Theory - Lecture 05
Let L contain K which contains F where all extensions are finite. In this video we prove [L:F] = [L:K][K:F]. This is a super useful formula.
From playlist Field Theory
This video is about polynomials with coefficients in a field.
From playlist Basics: Field Theory
Field Definition (expanded) - Abstract Algebra
The field is one of the key objects you will learn about in abstract algebra. Fields generalize the real numbers and complex numbers. They are sets with two operations that come with all the features you could wish for: commutativity, inverses, identities, associativity, and more. They
From playlist Abstract Algebra
Lecture 30. Fields, field extensions
0:00 Fields 1:48 Examples of fields 08:20 Characteristic of a field 11:20 Prime subfields (Q, F_p) 12:00 Every field has a prime subfield; relation of prime subfield to characteristic 20:15 Frobenius homomorphism 22:40 Field extension 23:50 A field extension of K possesses a structure of
From playlist Abstract Algebra 2
CTNT 2020 - Infinite Galois Theory (by Keith Conrad) - Lecture 1
The Connecticut Summer School in Number Theory (CTNT) is a summer school in number theory for advanced undergraduate and beginning graduate students, to be followed by a research conference. For more information and resources please visit: https://ctnt-summer.math.uconn.edu/
From playlist CTNT 2020 - Infinite Galois Theory (by Keith Conrad)
Visual Group Theory, Lecture 6.4: Galois groups
Visual Group Theory, Lecture 6.4: Galois groups The Galois group Gal(f(x)) of a polynomial f(x) is the automorphism group of its splitting field. The degree of a chain of field extensions satisfies a "tower law", analogous to the tower law for the index of a chain of subgroups. This hints
From playlist Visual Group Theory
Visual Group Theory, Lecture 6.5: Galois group actions and normal field extensions
Visual Group Theory, Lecture 6.5: Galois group actions and normal field extensions If f(x) has a root in an extension field F of Q, then any automorphism of F permutes the roots of f(x). This means that there is a group action of Gal(f(x)) on the roots of f(x), and this action has only on
From playlist Visual Group Theory
Álvaro Lozano-Robledo: Recent progress in the classification of torsion subgroups of...
Abstract: This talk will be a survey of recent results and methods used in the classification of torsion subgroups of elliptic curves over finite and infinite extensions of the rationals, and over function fields. Recording during the meeting "Diophantine Geometry" the May 22, 2018 at th
From playlist Math Talks
CTNT 2020 - Infinite Galois Theory (by Keith Conrad) - Lecture 4
The Connecticut Summer School in Number Theory (CTNT) is a summer school in number theory for advanced undergraduate and beginning graduate students, to be followed by a research conference. For more information and resources please visit: https://ctnt-summer.math.uconn.edu/
From playlist CTNT 2020 - Infinite Galois Theory (by Keith Conrad)
Basic/Primitive Extensions and Minimal Polynomials - Field Theory - Lecture 02
A "basic" or "primitive" extension of a field F is a new field F(alpha) where alpha in K an extension of F. We give some basic properties of extensions. Most importantly introduce the concept of minimal polynomials. @MatthewSalomone has some good videos on this already which might be mor
From playlist Field Theory
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)