& Give an example of a polynomial (x) € Z[] and a prime p such that f(x) is reducible in Qx] but 7(x) is irreducible in Z,JK]. Does this contradict
& Give an example of a polynomial (x) € Z[] and a prime p such that f(x) is reducible in Qx] but 7(x) is irreducible in Z,JK]. Does this contradict
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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#8 ignore the last part of the question.
![b My Questions | bartleby
Thomas W. Hungerford - Abstrac ×
+
O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(2014).pdf
-- A Read aloud V Draw
F Highlight
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141
of 621
(a) -x* +x +x+x+2
(c) 3x + 2 - 7x + 2x
(b) x* + 4x + x- x
(d) 2x - 5x+ 3x² + 4x – 6
(e) 2x + 7x + 5x² + 7x + 3
(f) 6x* – 31x + 25x2 + 33x + 7
2. Show that Vp is irrational for every positive prime integer p. [Hint: What are
the roots of x2 - p? Do you prefer this proof to the one in Exercises 30 and 31
of Section 1.37]
3. If a monic polynomial with integer coefficients has a root in Q, show that this
root must be an integer.
4. Show that each polynomial is irreducible in Q[x], as in Example 3.
(a) x* + 2x' +x + 1
(b) x* - 2x2 + 8x + 1
5. Use Eisenstein's Criterion to show that each polynomial is irreducible in Q[x]:
(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 + 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 + 6x? + 4x + 6
(Ъ) 9х + 4x3—3х + 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 ƒ(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
polynomials of degrees m and n, prove that it can be factored as a product of
monic polynomials of degrees m and n in Z[x].
B. 11. Prove that 30x – 91 (where n e Z, n>1) has no roots in Q.
12. Let Fbe a field and f(x) E F[×]. If c e Fand f(x +c) is irreducible in F[x],
prove that f(x) is irreducible in F[x]. [Hint: Prove the contrapositive.]
13. Prove that f(x) = x + 4x + 1 is irreducible in Q[x] by using Eisenstein's
Criterion to show that f(x + 1) is irreducible and applying Exercise 12.
14. Prove that f(x) = x* + x +²+x+lis irreducible in O[x]. [Hint: Use the
hint for Exercise 21 with p = 5.]
15. Let f(x) = a,t + an-1x1 +• ·•+ a,x + a, be a polynomial with integer
coefficients. If p is a prime such that p|a1, p|az, ... ,P|a, but p } m, and
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3:02 PM
O Search for anything
EPIC
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EPIC
11/20/2020](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27260fae-539c-4ca6-9fed-6022b8026087%2F21028c97-a14c-49f2-924a-7a2753f7a7d3%2F3iu5ujc_processed.png&w=3840&q=75)
Transcribed Image Text:b My Questions | bartleby
Thomas W. Hungerford - Abstrac ×
+
O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(2014).pdf
-- A Read aloud V Draw
F Highlight
O Erase
141
of 621
(a) -x* +x +x+x+2
(c) 3x + 2 - 7x + 2x
(b) x* + 4x + x- x
(d) 2x - 5x+ 3x² + 4x – 6
(e) 2x + 7x + 5x² + 7x + 3
(f) 6x* – 31x + 25x2 + 33x + 7
2. Show that Vp is irrational for every positive prime integer p. [Hint: What are
the roots of x2 - p? Do you prefer this proof to the one in Exercises 30 and 31
of Section 1.37]
3. If a monic polynomial with integer coefficients has a root in Q, show that this
root must be an integer.
4. Show that each polynomial is irreducible in Q[x], as in Example 3.
(a) x* + 2x' +x + 1
(b) x* - 2x2 + 8x + 1
5. Use Eisenstein's Criterion to show that each polynomial is irreducible in Q[x]:
(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 + 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 + 6x? + 4x + 6
(Ъ) 9х + 4x3—3х + 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 ƒ(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
polynomials of degrees m and n, prove that it can be factored as a product of
monic polynomials of degrees m and n in Z[x].
B. 11. Prove that 30x – 91 (where n e Z, n>1) has no roots in Q.
12. Let Fbe a field and f(x) E F[×]. If c e Fand f(x +c) is irreducible in F[x],
prove that f(x) is irreducible in F[x]. [Hint: Prove the contrapositive.]
13. Prove that f(x) = x + 4x + 1 is irreducible in Q[x] by using Eisenstein's
Criterion to show that f(x + 1) is irreducible and applying Exercise 12.
14. Prove that f(x) = x* + x +²+x+lis irreducible in O[x]. [Hint: Use the
hint for Exercise 21 with p = 5.]
15. Let f(x) = a,t + an-1x1 +• ·•+ a,x + a, be a polynomial with integer
coefficients. If p is a prime such that p|a1, p|az, ... ,P|a, but p } m, and
Cprt 2012 Cpe Leg AR Rig tand May aot be copind cand ard e wole ar la pet Detedarnie d, perty cot ay bepd mteBook endur Cn o. Edralview be
d atoy ppnd da at ty hat he ovnl ngpert Cgge Laming ma right omveddonl o any tme it dghta a uire
3:02 PM
O Search for anything
EPIC
Ai
EPIC
11/20/2020
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