(c) Let I = (f(x)) be the principal ideal generated by f(x) in Z2[r]. Calculate the multi- plicative inverse of (x³ + 1) + I in Z2[x]/I. (d) Use part (b) to prove that g(x) (1/6)r6 + (2/3)r³ + (4/3)æª + (7/6)x³ + (5/3)a² + (2/3)x + (5/6) is irreducible over Q.
(c) Let I = (f(x)) be the principal ideal generated by f(x) in Z2[r]. Calculate the multi- plicative inverse of (x³ + 1) + I in Z2[x]/I. (d) Use part (b) to prove that g(x) (1/6)r6 + (2/3)r³ + (4/3)æª + (7/6)x³ + (5/3)a² + (2/3)x + (5/6) is irreducible over Q.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Answer C and D only.
![Let f(x) = x® +x³ + 1.
(a) Prove that f(x) is reducible over Z3. Write f(x) as a product of irreducible factors.
(b) Prove that f (x) is irreducible over Z2.
(c) Let I = (f(x)) be the principal ideal generated by f(x) in Z2[x]. Calculate the multi-
plicative inverse of (x³ + 1) + I in Z2[x]/I.
%3D
(d) Use part (b) to prove that g(x)
(1/6)r6 + (2/3)³+ (4/3)aª + (7/6)x³ + (5/3)x² +
(2/3)x + (5/6) is irreducible over Q.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdd2e4611-cd8e-4235-a2f9-a8766d69ed68%2F47287218-eb00-44e3-a108-1f4ee9c55538%2F2e7zxk_processed.png&w=3840&q=75)
Transcribed Image Text:Let f(x) = x® +x³ + 1.
(a) Prove that f(x) is reducible over Z3. Write f(x) as a product of irreducible factors.
(b) Prove that f (x) is irreducible over Z2.
(c) Let I = (f(x)) be the principal ideal generated by f(x) in Z2[x]. Calculate the multi-
plicative inverse of (x³ + 1) + I in Z2[x]/I.
%3D
(d) Use part (b) to prove that g(x)
(1/6)r6 + (2/3)³+ (4/3)aª + (7/6)x³ + (5/3)x² +
(2/3)x + (5/6) is irreducible over Q.
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