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Unformatted text preview: f.x/ D a.x/b.x/ , where a.x/ D P r i D i x i and b.x/ D P s j D i x j . Since a D , exactly one of and is divisible by p ; suppose without loss of generality that p divides , and p does not divide . Considering the equations a 1 D 1 C 1 ::: a s 1 D s 1 C C s 1 a s D C s 1 1 C C s ; we obtain by induction that j is divisible by p for all j ( j s 1 ). Finally, the last equation yields that is divisible by p , a contradiction. n Example 6.8.5. x 3 C 14x C 7 is irreducible by the Eisenstein criterion. Example 6.8.6. Sometimes the Eisenstein criterion can be applied after a linear change of variables. For example, for the socalled cyclotomic polynomial f.x/ D x p 1 C x p 2 C C x 2 C x C 1;...
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 Fall '08
 EVERAGE
 Algebra, Factoring

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