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ANSWERS TO THE ODD-NUMBERED EXERCISES |
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7.39. |
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[ ] |
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7.41. |
s0 |
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7.43. |
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0, 1 |
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{[0], [1]}. {[0], [1], [10]}. 7.45. The monoid contains 27 elements.
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CF |
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Dormant R |
Dormant W |
Dormant W |
Buds |
F |
Dead |
1 |
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WCF |
R |
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RWC |
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RWCF |
CHAPTER 8 |
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8.1. |
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8.3. A ring.
8.5. Not a ring; not closed under multiplication. 8.7. Not a ring; not closed under addition.
8.9. A ring.
8.11. Not a ring; distributive laws do not hold. 8.17. A subring.
8.19. Not a subring; not closed under addition.
8.21. |
Neither. |
8.23. |
Both. |
8.25. |
Integral domain. |
8.29. |
[2], [4], [5], [6], [8]. |
ANSWERS TO THE ODD-NUMBERED EXERCISES |
315 |
8.31. Any nonempty proper subset of X. 8.33. Nonzero matrices with zero determinant.
8.37. f (x) = [x]6.
8.39. f (x, y) = (x, y), (y, x), (x, x) or (y, y).
8.47. (b) −1 and 0.
8.55.The identity is Dn(x) = (1/2π ) + (1/π )(cos x + cos 2x + · · · + cos nx). The ring is not an integral domain.
CHAPTER 9
9.1. |
23 x2 + 45 x − 158 and 458 x + 87 . |
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9.3. |
x4 + x3 + x2 + x and 1. |
and 1 − 2i, or 4 − 2i |
and −3 + i. |
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9.5. |
3 − i and 4 + 2i, or 4 − i |
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9.7. gcd(a, b) = 3, s = −5, t = 4. |
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9.9. gcd(a, b) = 1, s = −(2x + 1)/3, t = (2x + 2)/3. |
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9.11. gcd(a, b) = 2x + 1, s = 1, t = 2x + 1. |
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9.13. gcd(a, b) = 1, s = 1, t = −1 + 2i. |
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9.15. x = −6, y = 5. |
9.17. x = −14, y = 5. |
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9.19. [23]. |
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9.21. [17]. |
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9.23. |
No solutions. |
9.25. (x |
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9.29. x |
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9.27. |
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− 9x + 3. |
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9.33.(x − √2)(x + √2)(x − i√2)(x + i√2)(x − 1 − i)(x − 1 + i) (x + 1 − i)(x + 1 + i).9.31. x 4x 1.
9.35. (x2 − 2)(x2 + 2)(x2 − 2x + 2)(x2 + 2x + 2). |
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9.37. |
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9.39. x3 |
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9.41. Kerψ = {q(x) · |
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− 2x + 4)|q(x) Q[x]} and Im ψ |
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{a + b 3i|a, b Q}. |
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9.43. Irreducible by Eisenstein’s Criterion. |
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9.45. Irreducible, since it has no linear factors. |
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9.47. Reducible; any polynomial of degree >2 in R[x] is reducible. |
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9.49. No. |
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9.55. No. |
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9.61. x ≡ 40 mod 42. |
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9.63. x ≡ 22 mod 30. |
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9.67. |
65. |
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CHAPTER 10
10.1. ((0, 0)), ((0, 1)), ((1, 0)), Z2 × Z2. 10.3. (0) and Q.
316 |
ANSWERS TO THE ODD-NUMBERED EXERCISES |
10.5. (p(x)) where p(x) C[x].
10.7. The quotient ring is a field.
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10.9. The ideal ((1, 2)) is the whole ring Z3 × Z3. The quotient ring is not a field.
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((1, 2)) |
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((1, 2)) |
((1, 2)) |
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((1, 2)) |
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((1, 2)) |
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10.11. 8x + 2 and 14x + 97. |
10.13. x2 + x and x2. |
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10.17. (a) 6; (b) 36; (c) x2 − 1, (a) ∩ (b) = (lcm(a, b)). |
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10.33. No. |
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10.35. The whole ring. |
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10.37. |
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Z8 |
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([2]8) |
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([4]8) |
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([0]8) |
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10.39. Irreducible; Z11. |
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10.41. Reducible. |
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10.43. Irreducible; Q( |
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10.45. Irreducible; Q( 2, |
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10.47. Not a field; contains zero divisors. |
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10.49. A field by Corollary 10.16. |
10.51. A field by Theorem 10.17. |
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10.53. |
Not a field; x2 + 1 = (x + 2)(x + 3) in Z5[x]. |
10.55. |
A field isomorphic to Q[x]/(x4 − 11). |
10.59. |
(0) and (xn) for n 0; (x) is maximal. |