Abstract Algebra Theory and Applications - Computer Science ...

272 CHAPTER 15 POLYNOMIALS(c) p(x) = 4x 5 − x 3 + x 2 + 4 and q(x) = x 3 − 2 in Z 5 [x](d) p(x) = x 5 + x 3 − x 2 − x and q(x) = x 3 + x in Z 2 [x]4. Find the greatest common divisor of each of the following pairs p(x) and q(x)of polynomials. If d(x) = gcd(p(x), q(x)), find two polynomials a(x) and b(x)such that a(x)p(x) + b(x)q(x) = d(x).(a) p(x) = 7x 3 +6x 2 −8x+4 and q(x) = x 3 +x−2, where p(x), q(x) ∈ Q[x](b) p(x) = x 3 + x 2 − x + 1 and q(x) = x 3 + x − 1, where p(x), q(x) ∈ Z 2 [x](c) p(x) = x 3 +x 2 −4x+4 and q(x) = x 3 +3x−2, where p(x), q(x) ∈ Z 5 [x](d) p(x) = x 3 − 2x + 4 and q(x) = 4x 3 + x + 3, where p(x), q(x) ∈ Q[x]5. Find all of the zeros for each of the following polynomials.(a) 5x 3 + 4x 2 − x + 9 in Z 12(b) 3x 3 − 4x 2 − x + 4 in Z 5(c) 5x 4 + 2x 2 − 3 in Z 7(d) x 3 + x + 1 in Z 26. Find all of the units in Z[x].7. Find a unit p(x) in Z 4 [x] such that deg p(x) > 1.8. Which of the following polynomials are irreducible over Q[x]?(a) x 4 − 2x 3 + 2x 2 + x + 4(b) x 4 − 5x 3 + 3x − 2(c) 3x 5 − 4x 3 − 6x 2 + 6(d) 5x 5 − 6x 4 − 3x 2 + 9x − 159. Find all of the irreducible polynomials of degrees 2 and 3 in Z 2 [x].10. Give two different factorizations of x 2 + x + 8 in Z 10 [x].11. Prove or disprove: There exists a polynomial p(x) in Z 6 [x] of degree n withmore than n distinct zeros.12. If F is a field, show that F [x 1 , . . . , x n ] is an integral domain.13. Show that the division algorithm does not hold for Z[x]. Why does it fail?14. Prove or disprove: x p + a is irreducible for any a ∈ Z p , where p is prime.15. Let f(x) be irreducible. If f(x) | p(x)q(x), prove that either f(x) | p(x) orf(x) | q(x).16. Suppose that R and S are isomorphic rings. Prove that R[x] ∼ = S[x].17. Let F be a field and a ∈ F . If p(x) ∈ F [x], show that p(a) is the remainderobtained when p(x) is divided by x − a.18. Let Q ∗ be the multiplicative group of positive rational numbers. Prove thatQ ∗ is isomorphic to (Z[x], +).