18. Let p:C + C be an isomorphism of rings such that e(a) - a for each ae Q. Suppose r E Cisa root of fx) E Q[x]. Prove that p(r) is also a root of f(x).
18. Let p:C + C be an isomorphism of rings such that e(a) - a for each ae Q. Suppose r E Cisa root of fx) E Q[x]. Prove that p(r) is also a root of f(x).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Thomas W. Hungerford - Abstrac ×
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(b) If f(x), g(x) e F[x] have the same roots in F, are they associates in F[x]?
14. (a) Suppose r, s E Fare roots of ax? + bx +c€ F[x] (with a + 0p). Use the
Factor Theorem to show that r +s = -a-b and rs = a'c.
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4.4 Polynomial Functions, Roots, and Reducibility
111
(b) Suppose r, s, t e Fare roots of ax + bx² + cx + de F[x] (with a + Op).
Show that r + s+ t= -ab and rs + st + rt = acand rst = -a'd.
15. Prove that + 1 is reducible in Z[x] if and only if there exist integers a and b
such that p = a + b and ab = 1 (mod p).
16. Let f(x), g(x) e F[x] have degree sn and let co, C1, . .., C, be distinct elements
of F. If f(c) = g(c) for i = 0, 1, ..., n, prove that f(x) = g(x) in F[x].
17. Find a polynomial of degree 2 in Zx] that has four roots in Zg. Does this
contradict Corollary 4.17?
18. Let p:C → C be an isomorphism of rings such that o(a) = a for each
a E Q. Suppose re Cisa root of f(x) E Q[x]. Prove that o(r) is also a
root of f(x).
19. We say thata E Fis a multiple root of f(x) E F[x] if (x – a)k is a factor of
f(x) for some k 2 2.
(a) Prove that a e R is a multiple root of f(x) E R[x] if and only if a is a
root of both f(x) and f"(x), where f"(x) is the derivative of f(x).
(b) If f(x) E R[x] and if f(x) is relatively prime to f"(x), prove that f(x) has
no multiple root in R.
20. Let R be an integral domain. Then the Division Algorithm holds in R[x]
whenever the divisor is monic, by Exercise 14 in Section 4.1. Use this fact to
show that the Remainder and Factor Theorems hold in R[x].
21. If R is an integral domain and f(x) is a nonzero polynomial of degree n in
R[x], prove that f(x) has at most n roots in R. [Hint: Exercise 20.]
22. Show that Corollary 4.20 holds if Fis an infinite integral domain. [Hint: See
Exercise 21.]
23. Let f(x), g(x), h(x) e F[x] and re F.
(a) If f(x) = g(x) + h(x) in F[x], show that f(r) = g(r) + h(r) in F.
(b) If f(x) = g(x)h(x) in F[x], show that f(r) = g(r)h(r) in F.
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Thomas W. Hungerford - Abstrac ×
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O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(2014).pdf
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(b) If f(x), g(x) e F[x] have the same roots in F, are they associates in F[x]?
14. (a) Suppose r, s E Fare roots of ax? + bx +c€ F[x] (with a + 0p). Use the
Factor Theorem to show that r +s = -a-b and rs = a'c.
Capt 2012 C La ARig Ra d May act be pind caad or dy t in wale or in part Dee te eied,
d ttg d dday ate o ingpe Cleng dri
Bodk dr . Rartal v ba
datt
4.4 Polynomial Functions, Roots, and Reducibility
111
(b) Suppose r, s, t e Fare roots of ax + bx² + cx + de F[x] (with a + Op).
Show that r + s+ t= -ab and rs + st + rt = acand rst = -a'd.
15. Prove that + 1 is reducible in Z[x] if and only if there exist integers a and b
such that p = a + b and ab = 1 (mod p).
16. Let f(x), g(x) e F[x] have degree sn and let co, C1, . .., C, be distinct elements
of F. If f(c) = g(c) for i = 0, 1, ..., n, prove that f(x) = g(x) in F[x].
17. Find a polynomial of degree 2 in Zx] that has four roots in Zg. Does this
contradict Corollary 4.17?
18. Let p:C → C be an isomorphism of rings such that o(a) = a for each
a E Q. Suppose re Cisa root of f(x) E Q[x]. Prove that o(r) is also a
root of f(x).
19. We say thata E Fis a multiple root of f(x) E F[x] if (x – a)k is a factor of
f(x) for some k 2 2.
(a) Prove that a e R is a multiple root of f(x) E R[x] if and only if a is a
root of both f(x) and f"(x), where f"(x) is the derivative of f(x).
(b) If f(x) E R[x] and if f(x) is relatively prime to f"(x), prove that f(x) has
no multiple root in R.
20. Let R be an integral domain. Then the Division Algorithm holds in R[x]
whenever the divisor is monic, by Exercise 14 in Section 4.1. Use this fact to
show that the Remainder and Factor Theorems hold in R[x].
21. If R is an integral domain and f(x) is a nonzero polynomial of degree n in
R[x], prove that f(x) has at most n roots in R. [Hint: Exercise 20.]
22. Show that Corollary 4.20 holds if Fis an infinite integral domain. [Hint: See
Exercise 21.]
23. Let f(x), g(x), h(x) e F[x] and re F.
(a) If f(x) = g(x) + h(x) in F[x], show that f(r) = g(r) + h(r) in F.
(b) If f(x) = g(x)h(x) in F[x], show that f(r) = g(r)h(r) in F.
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