Complex Numbers (more)

1. Find the monic polynomial of smallest possible degree with real coefficients having 2 + i and Ö 5 - i as two of its roots.

2. Factor the polynomial x⁴ + 1 into a product of two second degree polynomials with real coefficients and also into a product of four first degree polynomials with complex coefficients.

3. Exercises: Section 4.3 (p. 186): 2acd, 4, 5a

4. Explain why each of the following polynomials is or is not irreducible over the rationals:

a. x³ + 4x² - 3x + 5

b. 4x⁴ - 6x² + 6x - 12

c. x²⁰⁰ - 1900

d. 4x⁸ + 4x⁴ + 1

e. 2x⁴ - 25x² - 5

f. x⁴ + x³ + x² + x + 1 (Hint: Let x = t + 1)

g. x⁸ - 256

h. x³ - 4x² + 7x - 12

5. Every commutative ring is a field.
6. Every field is a commutative ring.
7. Every reducible polynomial of degree 4 over Q has a root in Q.
8. If a polynomial has a root in a field K then it is not irreducible.
9. Every polynomial that is reducible over Z is also reducible over Q.
10. Every polynomial that is reducible over Q is also reducible over Z.
11. Every polynomial of odd degree over R is reducible over R.
12. Every polynomial of odd degree over R is reducible over Q.
13. Every polynomial of prime degree is irreducible over Q.
14. Every polynomial of composite degree is reducible over Q.
15. All second degree polynomials over R have a root in R.
16. All second degree polynomials over R have a root in C.
17. All first degree polynomials over any field K are irreducible.
18. All third degree polynomials over R must have a root in R.
19. Z_m is a field for all positive integers m.
20. For all positive integers m and all polynomials in Z_m[x], the degree of the sum of any two polynomials is the sum of the degrees.