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 ˇ i x j . Since a D ˛ ˇ , exactly one of ˛ and ˇ is divisible by p ; suppose without loss of generality that p divides ˇ , and p does not divide ˛ . Considering the equations a 1 D ˇ 1 ˛ C ˇ ˛ 1 ::: a s ± 1 D ˇ s ± 1 ˛ C ²²² C ˇ ˛ s ± 1 a s D ˛ C ˇ s ± 1 ˛ 1 C ²²² C ˇ ˛ s ; we obtain by induction that ˇ j is divisible by p for all j ( ³ j ³ 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 so–called cyclotomic polynomial f.x/ D x p ± 1 C x p ± 2 C ²²² C x 2 C x C 1;...
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 Fall '08
 EVERAGE
 Algebra, Factoring, Prime number, sufficient condition, 4 g, Eisenstein, Eisenstein criterion

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