7. Show that each polynomial f(x) such that f(x) is irreducible in Z, (a) 7x + 6x? + 4x + 6 (b)
7. Show that each polynomial f(x) such that f(x) is irreducible in Z, (a) 7x + 6x? + 4x + 6 (b)
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 x
b My Questions | bartleby
O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(201..
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J. Use EISENStein s Citerion to show that each polynomial is irreaUcIDle in wX]:
(a) x – 4x + 22
(b) 10 – 15x + 25x² – 7x*
(c) 5x" – 6x* + 12x + 36x – 6
6. Show that there are infinitely many integers k such that x + 12x – 21x + k
is irreducible in Q[x].
7. Show that each polynomial f(x) is irreducible in Q[x] by finding a prime p
such that f(x) is irreducible in Z,[x]
(a) 7x + 6x² + 4x + 6
(b) 9x + 4x – 3x + 7
8. Give an example of a polynomial f(x) E Z[x] and a prime p such that f(x)
is reducible in Q[x] but f(x) is irreducible in Z,[x]. Does this contradict
Theorem 4.25?
9. Give an example of a polynomial in Z[x] that is irreducible in Q[x] but factors
when reduced mod 2, 3, 4, and 5.
10. If a monic polynomial with integer coefficients factors in Z[x] as a product of
11:05 AM
O Type here to search
口
EPIC
Ai
EPIC
50
12/11/2020](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27260fae-539c-4ca6-9fed-6022b8026087%2Fe21e4d48-11af-4677-9fb2-a97112af8f3a%2F7q80od_processed.png&w=3840&q=75)
Transcribed Image Text:Thomas W. Hungerford - Abstrac x
b My Questions | bartleby
O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(201..
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of 621
+
--| A' Read aloud
V Draw
F Highlight
O Erase
141
J. Use EISENStein s Citerion to show that each polynomial is irreaUcIDle in wX]:
(a) x – 4x + 22
(b) 10 – 15x + 25x² – 7x*
(c) 5x" – 6x* + 12x + 36x – 6
6. Show that there are infinitely many integers k such that x + 12x – 21x + k
is irreducible in Q[x].
7. Show that each polynomial f(x) is irreducible in Q[x] by finding a prime p
such that f(x) is irreducible in Z,[x]
(a) 7x + 6x² + 4x + 6
(b) 9x + 4x – 3x + 7
8. Give an example of a polynomial f(x) E Z[x] and a prime p such that f(x)
is reducible in Q[x] but f(x) is irreducible in Z,[x]. Does this contradict
Theorem 4.25?
9. Give an example of a polynomial in Z[x] that is irreducible in Q[x] but factors
when reduced mod 2, 3, 4, and 5.
10. If a monic polynomial with integer coefficients factors in Z[x] as a product of
11:05 AM
O Type here to search
口
EPIC
Ai
EPIC
50
12/11/2020
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