College Algebra Exam Review 304

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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;...
View Full Document

Ask a homework question - tutors are online