(c) x + 4x + 2x + 3x -x+5 (4) x + Sx + 4x + 7 19. Write euch polynomial as a product of irreducible polynomiałs in Q[3).

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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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
142
of 621
T2. Let F be a neid and jX) E FXI. II CE Fand JX + C) IS Irreaucibie in F|X],
prove that f(x) is irreducible in F[x]. [Hint: Prove the contrapositive.]
13. Prove that f(x) = xt + 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 + + x + l is 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
Ot 2012 C Le A Rig taved May at be opind cd ar we ar la pt Dete drie d perty cot y bepd me Bootendtr C o. Edalvew ba
dd t oy ppd t dan at ty het he ovlrgpert Cgge Laig ma right tomveddonl at y tme i dghs anmguire
120 Chapter 4 Arithmetic in F[x]
pta, prove that f(x) is irreducible in Q[x). [Hint: Let y = 1/x in f(x)/x"; the
resulting polynomial is irreducible, by Thcorem 4.24.]
16. Show by example that this statement is false: If f(x) E 7[x] and there is no
prime p satisfying the hypotheses of Theorem 4.24, then f(x) is reducible in Qx].
17. Show that there are n+1 - n polynomials of degree k in Z[x].
18. Which of these polynomials are irreducible in Q[x]:
(a) メ-+1
(c) x + 4x + 2x+ 3x - x + 5
(b) x* +x +1
(d) x + 5x + 4x + 7
19. Write each polynomial as a product of irreducible polynomials in Q[x].
(a) + 2x – 6x? – 16x – 8
() x" – 2x – 6x– 15x – 33x – 9
20. If f(x) = a, +..+ a,x + ao, 8(x) = b,x + ...+ b,x + bo, and h(x) =
cxt + ...+ cx + co are polynomials in Z[x] such that f(x) = g(x)h(x), show
that in Z[x], F(x) = (x). Also, see Exercise 19 in Section 4.1.
C.21. Prove that for p prime, f(x) = x-1+ -2+ . ..+ x² + x + 1 is irreducible
in Q[x]. [Hint: (x – 1)f(x) = x – 1, so that f(x) = (x – 1)/(x – 1) and
f(x + 1) = [(x + 1y –11/x. Expand (x + 1)® by the Binomial Theorem
CoursSa
(Appendix E) and note that p divides
when k > 0. Use Eisenstein's
Criterion to show that (x + 1) is irreducible; apply Exercise 12.]
EXCURSION: Geometric Constructions (Chapter 15) may be covered at
this point if desired.
4.6
Irreducibility in R[x] and C[x]*
3:03 PM
O Search for anything
口
EPIC
Ai
へ
EPIC
11/20/2020
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 142 of 621 T2. Let F be a neid and jX) E FXI. II CE Fand JX + C) IS Irreaucibie in F|X], prove that f(x) is irreducible in F[x]. [Hint: Prove the contrapositive.] 13. Prove that f(x) = xt + 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 + + x + l is 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 Ot 2012 C Le A Rig taved May at be opind cd ar we ar la pt Dete drie d perty cot y bepd me Bootendtr C o. Edalvew ba dd t oy ppd t dan at ty het he ovlrgpert Cgge Laig ma right tomveddonl at y tme i dghs anmguire 120 Chapter 4 Arithmetic in F[x] pta, prove that f(x) is irreducible in Q[x). [Hint: Let y = 1/x in f(x)/x"; the resulting polynomial is irreducible, by Thcorem 4.24.] 16. Show by example that this statement is false: If f(x) E 7[x] and there is no prime p satisfying the hypotheses of Theorem 4.24, then f(x) is reducible in Qx]. 17. Show that there are n+1 - n polynomials of degree k in Z[x]. 18. Which of these polynomials are irreducible in Q[x]: (a) メ-+1 (c) x + 4x + 2x+ 3x - x + 5 (b) x* +x +1 (d) x + 5x + 4x + 7 19. Write each polynomial as a product of irreducible polynomials in Q[x]. (a) + 2x – 6x? – 16x – 8 () x" – 2x – 6x– 15x – 33x – 9 20. If f(x) = a, +..+ a,x + ao, 8(x) = b,x + ...+ b,x + bo, and h(x) = cxt + ...+ cx + co are polynomials in Z[x] such that f(x) = g(x)h(x), show that in Z[x], F(x) = (x). Also, see Exercise 19 in Section 4.1. C.21. Prove that for p prime, f(x) = x-1+ -2+ . ..+ x² + x + 1 is irreducible in Q[x]. [Hint: (x – 1)f(x) = x – 1, so that f(x) = (x – 1)/(x – 1) and f(x + 1) = [(x + 1y –11/x. Expand (x + 1)® by the Binomial Theorem CoursSa (Appendix E) and note that p divides when k > 0. Use Eisenstein's Criterion to show that (x + 1) is irreducible; apply Exercise 12.] EXCURSION: Geometric Constructions (Chapter 15) may be covered at this point if desired. 4.6 Irreducibility in R[x] and C[x]* 3:03 PM O Search for anything 口 EPIC Ai へ EPIC 11/20/2020
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