Let F be a field and let a be a nonzero element of F. (a) If af(x) is irreducible over F, prove that f(x) is irreducible over F. (b) If f(x + a) is irreducible over F, prove that f(x) is irreducible of over F. (c) Use part b) to prove that f (r) = 8x3 – 6x +1 is irreducible over Q. Hint: Consider f(r +1).

Let F be a field and let a be a nonzero element of F. (a) If af(x) is irreducible over F, prove that f(x) is irreducible over F. (b) If f(x + a) is irreducible over F, prove that f(x) is irreducible of over F. (c) Use part b) to prove that f (r) = 8x3 – 6x +1 is irreducible over Q. Hint: Consider f(r +1).

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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