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(ax) is irreducible over F, prove that f(x) is irreducible over F.c. If f(x + a) is irreducible over F, prove that f(x) is irreducibleover F.d. Use part c to prove that 8x3- 6x + 1 is irreducible over Q.
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(ax) is irreducible over F, prove that f(x) is irreducible over F.c. If f(x + a) is irreducible over F, prove that f(x) is irreducibleover F.d. Use part c to prove that 8x3- 6x + 1 is irreducible over Q.
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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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(ax) is irreducible over F, prove that f(x) is irreducible over F.
c. If f(x + a) is irreducible over F, prove that f(x) is irreducible
over F.
d. Use part c to prove that 8x3- 6x + 1 is irreducible over Q.
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