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Unformatted text preview: polynomial of degree n has no rational root, but for some prime p its reduction mod p has irreducible factors of degrees 1 and n ± 1 , then then f is irreducible (Exercise 6.8.1 ). Exercises 6.8 6.8.1. Show that if a polynomial f.x/ 2 Z ŒxŁ of degree n has no rational root, but for some prime p its reduction mod p has irreducible factors of degrees 1 and n ± 1 , then then f is irreducible. 6.8.2. Provide the details for Example 6.8.6 . 6.8.3. Show that for each natural number n .x ± 1/.x ± 2/.x ± 3/ ²²² .x ± n/ ± 1 is irreducible over the rationals. 6.8.4. Show that for each natural number n ¤ 4 .x ± 1/.x ± 2/.x ± 3/ ²²² .x ± n/ C 1 is irreducible over the rationals....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra

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