Field Theory (Abstract Algebra)
55 videos • 70,991 views • by Nani's MathsClass
In this course, we shall be studying the Field theory and its extensions. Course Instructor: Meenal Singhal
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#01 Field Theory : Motivation and Basics
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#02 Field Theory : Definition of Field
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#03 Field Theory : Field of Rationals, Reals and Complex numbers
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#04 Field Theory : Example of a field
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#05 Field Theory : Integer modulus p (p is prime) is a field.
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#06 Field Theory : What is a Subfield?
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#07 Field Theory : Results on Subfield
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#08 Field Theory : Finite field and its examples
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#09 Field Theory : Characteristic of a field
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#10 Field Theory : ch(F) is either 0 or a prime number
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11
#11 Field Theory : What is a prime subfield?
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#12 Field Theory : Order of a finite field
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#13 Field Theory : Revisiting polynomials
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#14 Field Theory : It all starts with polynomials!
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15
#15 Field Theory : Order of a subfield of a finite field
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#16 Field Theory : Reducibility tests for polynomials of degree 2 or 3
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#17 Field Theory : Irreducibility tests for polynomials
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#18 Field Theory : Eisentein's Criterion
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#19 Field Theory : Cyclotomic polynomials
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#20 Field Theory : Connection between irreducible polynomials and Field
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#21 Field Theory : What is a field extension?
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#22 Field Theory : Finite extension of a finite extension is a finite extension
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#23 Field Theory : Embedding of a field F into a field E implies E is an extension of F
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#24 Field Theory : Kronecker Theorem (Existence of extension)
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#25 Field Theory : How to create an extension field by adjoining an element?
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#26 Field Theory : Algebraic and Transcendental Field Extensions
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#27 Field Theory : Every finite extension is an algebraic extension
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#28 Field Theory : Not every Algebraic extension is a field extension
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#29 Field Theory : What is a splitting field?-I
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#30 Field Theory : What is a splitting field?-II
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#31 Field Theory : Existence of splitting field?
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#32 Field Theory : Uniqueness of splitting field?
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#33 Field Theory : A splitting field implies a finite and an algebraic extension
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#34 Field Theory : Example of a splitting field
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#35 Field Theory : What is a normal extension?
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#36 Field Theory : Equivalent statements for a normal extension
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#37 Field Theory : Finite extension is normal extension if and only if it is a splitting field
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#38 Field Theory : An extension field with degree 2 is a normal extension
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#39 Field Theory: Multiple roots of an irreducible polynomial in an extension field
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#40 Field Theory: An irreducible polynomial has multiple roots iff f’(x)=0
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41
#41 Field Theory: An irreducible polynomial over F has simple, multiple roots if char(F)=0, p resp.
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#42 Field Theory: Multiplicity of any irreducible polynomial over any field
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#43 Field Theory: What is a prime field?
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#44 Field Theory: The prime field is either isomorphic to Q or to Z/(p), where p is prime
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#45 Field Theory: A finite field F has sub field isomorphic to Z/(p) and has p^n elements
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#46 Field Theory: The roots of x^(p^n)-x in Zp[x] are all distinct and form splitting field over Zp
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#47 Field Theory: Any finite field has an extension of any given finite degree
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#48 Field Theory: The multiplicative group of non-zero elements of a finite field is cyclic
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#49 Field Theory: What are separable extension, simple extension and perfect fields?
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#50 Field Theory: A finite and separable extension is also a simple extension
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#51 Field Theory: Steinitz Theorem, A NASC for a finite extension to be simple
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#52 Field Theory: What is an automorphism group?
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#53 Field Theory: A fixed field E_H is an intermediate field between E and F
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#54 Field Theory: Example 1 of automorphism group and fixed field
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#55 Field Theory: Example 2 of automorphism group and fixed field
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