ASL STEM

Galois Extension Field Sign Video

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Definition

The following are equivalent definitions for a Galois extension field (also simply known as a Galois extension) K of F. 1. K is the splitting field for a collection of separable polynomials. When K is a finite extension, then only one separable polynomial is necessary. 2. The field automorphisms of K that fix F do not fix any intermediate fields E, i.e., F subset E subset K. 3. Every irreducible polynomial over F which has a root in K factors into linear factors in K. Also, K must be a separable extension. 4. A field automorphism sigma:F* -> F* of the algebraic closure F* of F for which sigma(K)=K must fix F. That is to say that sigma must be a field automorphism of K fixing F. Also, K must be a separable extension.

Source: http://mathworld.wolfram.com

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+ Characteristic

+ Symmetric Group

+ Cancellation Law *

+ Trivial Subgroup *

+ General Linear Group *

+ Induced Law *

+ Lagrange's Theorem *

+ Special Linear Group *

+ Proper Subgroup *

+ Law Of Composition *

+ Product Set *

+ Transposition *

+ Abelian Group

+ Infinite Cyclic *

+ Quaternion Group *

+ Klein Four Group *

+ Cyclic Subgroup

+ Skew Symmetric *

+ Trivial Homomorphism *

+ Inclusion Map *

+ Sign Homomorphism *

+ Alternating Group *

+ Normal Subgroup

+ Isomorphism Class *

+ Automorphism *

+ Right Coset *

+ Subgroup Index *

+ Correspondence Theorem *

+ Counting Formula *

+ Conjugate Subgroup *

+ Group Operation *

+ Product Group *

+ Quotient Group *

+ Canonical Map *

+ Permutation Representation *

+ Cayley's Theorem *

+ Conjugacy Class *

+ Centralizer *

+ Class Equation *

+ Self Adjoint *

+ Subdiagonal *

+ Superdiagonal *

+ Commutator Subgroup *

+ Fixed Vector *

+ First Isomorphism Theorem *

+ Proper Subspace *

+ Prime Field *

+ Characteristic Zero *

+ Primitive Root *

+ Jordan Block *

+ Unitary Group *

+ Hypervector *

+ Linear Relation *

+ Basechange Matrix *

+ Dimension Formula *

+ Jordan Canonical Form *

+ Jordan Decomposition Theorem *

+ Kronecker Delta *

+ Orthogonal Group *

+ Special Orthogonal Group *

+ Orthogonal Operator *

+ Axis Of Rotation *

+ Rotation Matrix *

+ Rotation Group *

+ Cayley Hamilton Theorem *

+ Matrix Exponential *

+ Unitary Matrix *

+ First Sylow Theorem *

+ Linear Group *

+ Dihedral Group *

+ Translation Group *

+ Discrete Subgroup *

+ Representation *

+ Point Group *

+ Tetrahedral Group *

+ Octahedral Group *

+ Icosahedral Group *

+ Truncated Polyhedron *

+ Sylow P Subgroup *

+ Second Sylow Theorem *

+ Third Sylow Theorem *

+ Bilinear Form *

+ Positive Definite *

+ Negative Semidefinite *

+ Negative Definite *

+ Hermitian Form *

+ Standard Hermitian Form *

+ Hermitian Symmetry *

+ Positive Semidefinite *

+ Nondegenerate *

+ Orthogonal Sum *

+ Orthogonal Projection *

+ Hermitian Space *

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+ Lorentz Group *

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+ Gauss's Lemma *

+ Sieve Of Eratosthenes *

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+ Product Ideal *

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+ Complete Set *

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+ Quotient Ring *

+ Ascending Chain Condition *

+ Noetherian Ring *

+ Hilbert Basis Theorem *

+ Rational Canonical Form *

+ Extension Field *

+ Algebraic Number Field *

+ Finite Field *

+ Function Field *

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+ Primitive Element *

+ Algebraically Closed *

+ Symmetric Polynomial *

+ Sylvester's Inertia Law *

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+ Projective Group *

+ Elementary Symmetric Functions *

+ Symmetric Functions Theorem *

+ Discriminant *

+ Tschirnhausen Transformation *

+ Splitting Field *

+ Splitting Theorem *

+ Galois Group *

- Galois Extension Field *

+ Fixed Field *

+ Fixed Field Theorem *

+ Luroth's Theorem *

+ Intermediate Field *

+ Crystallographic Point Group *

+ Free Semigroup *

+ Resolvent Cubic *

+ Cyclotomic Field *

+ Kronecker Weber Theorem *

+ Kummer Extension *

+ Linear Operator *

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+ Commutative Ring

+ Generalized Eigenvector *

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