College Algebra Exam Review 304

College Algebra Exam Review 304 - f.x D a.x/b.x where a.x D...

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314 6. RINGS a prime p and factoring the reduction. Unfortunately, the condition of the proposition is merely a sufficient condition. It is quite possible (but rare) for a polynomial to be irreducible over Q but nevertheless for its reductions modulo every prime to be reducible. Example 6.8.2. Let f.x/ D 83 C 82x ± 99x 2 ± 87x 3 ± 17x 4 . The reduction of f modulo 3 is f 2 C x C x 4 g , which is irreducible over Z 3 . Hence, f is irreducible over Q . Example 6.8.3. Let f.x/ D ± 91 ± 63x ± 73x 2 C 22x 3 C 50x 4 . The reduction of f modulo 17 is 16 ± 6 C 12x C 5x 2 C 12x 3 C x 4 ² , which is irreducible over Z 17 . Therefore, f is irreducible over Q . A related sufficient condition for irreducibility is Eisenstein’s criterion: Proposition 6.8.4. (Eisenstein’s criterion). Consider a monic polynomial f.x/ D x n C a n ± 1 x n ± 1 C²²²C a 1 x C a 0 with integer coefficients. Suppose that p is a prime that divides all the coefficients a i and such that p 2 does not divide a 0 . Then f.x/ is irreducible over Q . Proof. If f has a proper factorization over Q , then it also has a proper factorization over Z , with both factors monic polynomials. Write
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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 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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