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11. Let f (x) E Z[x] with deg (f(x)) > 1. Let f'(x) be the polynomial in Zp,p € Z, obtained from f(x) by reducing all the coefficients of f(x) modulo p. Assume that deg (f(x)) = deg(f(x)), then: a) If f(x) is reducible over Zp, then f(x) is reducible over Q b) If f(x) is irreducible over Zp, then f(x) is irreducible over Q c) If f(x) is reducible over Q, then f(x) is irreducible over Z, d) None of the above

11. Let f (x) E Z[x] with deg (f(x)) > 1. Let f'(x) be the polynomial in Zp,p € Z, obtained from f(x) by reducing all the coefficients of f(x) modulo p. Assume that deg (f(x)) = deg(f(x)), then: a) If f(x) is reducible over Zp, then f(x) is reducible over Q b) If f(x) is irreducible over Zp, then f(x) is irreducible over Q c) If f(x) is reducible over Q, then f(x) is irreducible over Z, d) None of the above

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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11. Let f (x) E Z[x] with deg (f(x)) 2 1. Let f (x) be the polynomial in Zp, p € Z, obtained from f(x) by reducing all the coefficients of f (x) modulo p. Assume that deg (f(x)) = deg(f(x)), then: a) If f(x) is reducible over Zp, then f(x) is reducible over Q b) If f(x) is irreducible over Zp, then f(x) is irreducible over Q c) If f(x) is reducible over Q, then f(x) is irreducible over Z, d) None of the above
Transcribed Image Text:11. Let f (x) E Z[x] with deg (f(x)) 2 1. Let f (x) be the polynomial in Zp, p € Z, obtained from f(x) by reducing all the coefficients of f (x) modulo p. Assume that deg (f(x)) = deg(f(x)), then: a) If f(x) is reducible over Zp, then f(x) is reducible over Q b) If f(x) is irreducible over Zp, then f(x) is irreducible over Q c) If f(x) is reducible over Q, then f(x) is irreducible over Z, d) None of the above
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