Let E/F be a field extension and a∈E be algebraic over F. Suppose f∈F[X] is irreducible and degree of f is greater than or equal to 2, also gcd(deg(f),[F(a):F])=1. Prove that f has no roots in F(a).

Let E/F be a field extension and a∈E be algebraic over F. Suppose f∈F[X] is irreducible and degree of f is greater than or equal to 2, also gcd(deg(f),[F(a):F])=1. Prove that f has no roots in F(a).

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 8E: Let be an irreducible polynomial over a field . Prove that is irreducible over for all nonzero ...

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1. Let E/F be a field extension and a∈E be algebraic over F. Suppose f∈F[X] is irreducible and degree of f is greater than or equal to 2, also gcd(deg(f),[F(a):F])=1. Prove that f has no roots in F(a). Please do fast asap

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