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Galois Extension Field

  • 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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