Introduction to Abstract Algebra, Solutions Manual, 4th Edition

4.2 Factorization of Polynomials over a Field

1. a. f = a(a⁻¹f).

2. a. Yes, since a ≠ 0, f(b) = 0if and only if af(b) = 0.

3. a. f(1) = 0; indeed f = (x − 1)(x² − x + 2) over any field.

4.

a. Irreducible because it has no roots in

c. in . Not irreducible.

e. Irreducible, because it has no root in

5.

^∗Every polynomial of odd degree in has a root in –see Exercise 9.

7. f = [x − (1 − i)][x − (1 + i)][x − i][x + i] = (x² − 2x + 2)(x² + 1) = x⁴ − 2x³ + 3x² − 2x + 2.

The polynomial f² has the same roots, albeit of different multiplicities.

8. a. As f is monic, we may assume that both factors are monic (Exercise 6). Hence Now equate coefficients.

9. Assume f = a_nxⁿ + a_n−1xⁿ⁻¹ + ...