(a) Let φ:C → C be an isomorphism such that, for every a ∈ Q, φ(a) = a. Let z ∈ C be a root of f(X) ∈ Q[X]. Prove that φ(z) is also a root of f(X).(b) Let Φ : F[X] → F[X] be an isomorphism such that Φ(a) = a for every a ∈ F. Prove that f ∈ F[X] is irreducible if and only if Φ(f) is irreducible. Give an example of an isomorphism Φ : F[X] → F[X] such that Φ(a) = a for every a ∈ F but φ is not the identity.

(a) Let φ:C → C be an isomorphism such that, for every a ∈ Q, φ(a) = a. Let z ∈ C be a root of f(X) ∈ Q[X]. Prove that φ(z) is also a root of f(X).(b) Let Φ : F[X] → F[X] be an isomorphism such that Φ(a) = a for every a ∈ F. Prove that f ∈ F[X] is irreducible if and only if Φ(f) is irreducible. Give an example of an isomorphism Φ : F[X] → F[X] such that Φ(a) = a for every a ∈ F but φ is not the identity.

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 16E

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