Let K be the splitting field of f(x) = F(x)and let p(x) be an irreducible factor off(x) in F(x). If the roots of p(x) areα₁, α2,... O₁, then for each i there existsan automorphism o¡ in G(K, F) such that0₁(α₁) = 0₁.

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Answered: Let K be the splitting field of f(x) =…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.3: Factorization In F [x]

Problem 5E: Let F be a field and f(x)=a0+a1x+...+anxnF[x]. Prove that x1 is a factor of f(x) if and only if...

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Let K be the splitting field of f(x) = F(x) and let p(x) be an irreducible factor of f(x) in F(x). If the roots of p(x) are ¤₁, α2,... Oį, then for each i there exists an automorphism o; in G(K, F) such that 1₁(α₁) = 0₁.