(a) Assume p is prime. Show that there are (p- 1)/2 irreducible polynomials of the form f(x) = x² – b in Z,[r]. (b) Show that for every prime p, there exists a field with p2 elements.
(a) Assume p is prime. Show that there are (p- 1)/2 irreducible polynomials of the form f(x) = x² – b in Z,[r]. (b) Show that for every prime p, there exists a field with p2 elements.
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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![Exercise 6.3.13
(a) Assume p is prime. Show that there are (p - 1)/2 irreducible polynomials of the form
f(x) = x – b in Z,[r].
(b) Show that for every prime p, there exists a field with p2 elements.
There is actually a formula for the number of irreducible polynomials of degree d over
Z, or any finite field. See Dornhoff and Hohn [25, p. 377].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77285bc8-0a2e-4167-8a0c-57e44c45defb%2Fd100d7f0-80b8-4695-85ea-1987b72f1dda%2Fbk8lmjb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 6.3.13
(a) Assume p is prime. Show that there are (p - 1)/2 irreducible polynomials of the form
f(x) = x – b in Z,[r].
(b) Show that for every prime p, there exists a field with p2 elements.
There is actually a formula for the number of irreducible polynomials of degree d over
Z, or any finite field. See Dornhoff and Hohn [25, p. 377].
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