(i) If m / ±1 is a square-free integer prove Xn-m is irreducible in Q[X]. (ii) Prove that for every prime integer p the polynomial XP-1+XP-2+...+X+1 is irreducible in Q [X]. (iii) Prove that X4 - X-4 is irreducible in Z[X]. 8.
(i) If m / ±1 is a square-free integer prove Xn-m is irreducible in Q[X]. (ii) Prove that for every prime integer p the polynomial XP-1+XP-2+...+X+1 is irreducible in Q [X]. (iii) Prove that X4 - X-4 is irreducible in Z[X]. 8.
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. (i) If m / ±1 is a square-free integer prove Xn-m is irreducible in Q[X].
(ii) Prove that for every prime integer p the polynomial XP-1 + XP-2+...+X+1 is irreducible
in Q [X].
(iii) Prove that X4 - X - 4 is irreducible in Z[X].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff7373726-0d35-46ca-aa27-97bcf9126f3a%2F05c3bc85-07ff-4fe9-8921-a5c752293b9d%2Fe6egpk7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:8. (i) If m / ±1 is a square-free integer prove Xn-m is irreducible in Q[X].
(ii) Prove that for every prime integer p the polynomial XP-1 + XP-2+...+X+1 is irreducible
in Q [X].
(iii) Prove that X4 - X - 4 is irreducible in Z[X].
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