Answered: Let F be a field and let p(x), a1(x),…

Let F be a field and let p(x), a1(x), a2(x), . . . , ak(x) ∈ F[x], wherep(x) is irreducible over F. If p(x) | a1(x)a2(x) ... ak(x), show thatp(x) divides some ai(x).

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.4: Zeros Of A Polynomial

Problem 33E: Let where is a field and let . Prove that if is irreducible over , then is irreducible over .

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Question

Let F be a field and let p(x), a₁(x), a₂(x), . . . , a_k(x) ∈ F[x], where
p(x) is irreducible over F. If p(x) | a₁(x)a₂(x) ... a_k(x), show that
p(x) divides some a_i(x).

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