Chapter 8.3, Problem 2E

Let Q denote the field of rational numbers, R the field of real numbers, and C the field of complex. Determine whether each of the following polynomials is irreducible over each of the indicated fields, and state all the zeroes in each of the fields.

a. $x^2-2$ over Q, R, and C

b. $x^2+1$ over Q, R, and C

c. $x^2+x-2$ over Q, R, and C

d. $x^2+2x+2$ over Q, R, and C

e. $x^2+x+2$ over ${\text{Z}}_3$, ${\text{Z}}_5$, and ${\text{Z}}_7$

f. $x^2+2x+2$ over ${\text{Z}}_3$, ${\text{Z}}_5$, and ${\text{Z}}_7$

g. $x^3-x^2+2x+2$ over ${\text{Z}}_3$, ${\text{Z}}_5$, and ${\text{Z}}_7$

h. $x^4+2x^2+1$ over ${\text{Z}}_3$, ${\text{Z}}_5$, and ${\text{Z}}_7$