3. Suppose p is prime and E is a field extension of Zp. Suppose there is a € E which is a zero of rP – x+ 1. (a) Prove that xP – x + 1 = (x – a)· …· (x – a – p + 1). (b) Prove that ma,z,(x) = xº – x + 1. (Hint. Use part (a) and ma̟z,(x)|xP – x + 1.) (c) Deduce that xP – x +1 is irreducible in Z,[r].
3. Suppose p is prime and E is a field extension of Zp. Suppose there is a € E which is a zero of rP – x+ 1. (a) Prove that xP – x + 1 = (x – a)· …· (x – a – p + 1). (b) Prove that ma,z,(x) = xº – x + 1. (Hint. Use part (a) and ma̟z,(x)|xP – x + 1.) (c) Deduce that xP – x +1 is irreducible in Z,[r].
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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![3. Suppose p is prime and E is a field extension of Zp. Suppose there is a e E which is a zero of
xP – x + 1.
(a) Prove that xP – x +1 = (x – a) · · · (x – a – p+ 1).
(b) Prove that ma,z,(x) = xP – x+1. (Hint. Use part (a) and ma,Z,(x)|xP – x + 1.)
(c) Deduce that æP – x +1 is irreducible in Zp[r].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F833481a2-df8c-4805-95a2-f24b64ba619f%2F8cdd4ca8-0b5f-4aec-953c-4c68b37d3a03%2Fja7bzn_processed.png&w=3840&q=75)
Transcribed Image Text:3. Suppose p is prime and E is a field extension of Zp. Suppose there is a e E which is a zero of
xP – x + 1.
(a) Prove that xP – x +1 = (x – a) · · · (x – a – p+ 1).
(b) Prove that ma,z,(x) = xP – x+1. (Hint. Use part (a) and ma,Z,(x)|xP – x + 1.)
(c) Deduce that æP – x +1 is irreducible in Zp[r].
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