2. Let p be a prime and let a € F,- {0}. = (a) Show that there exists a natural number mEN such that a" 1. [Hint: first show that the set {a: k€ N} is finite.] (b) The order of a, denoted ord(a), is defined to be the smallest natural number n such that a" = 1. What are the orders of the elements of F,- {0} for p = 3,5,7. (c) Show that if a = 1 for some m € Z then n = ord(a) divides m. [Hint: show that ged(m, n) = n.] (d) Show that ord(a) divides p-1. [Hint: you may use Fermat's Little Theorem from
2. Let p be a prime and let a € F,- {0}. = (a) Show that there exists a natural number mEN such that a" 1. [Hint: first show that the set {a: k€ N} is finite.] (b) The order of a, denoted ord(a), is defined to be the smallest natural number n such that a" = 1. What are the orders of the elements of F,- {0} for p = 3,5,7. (c) Show that if a = 1 for some m € Z then n = ord(a) divides m. [Hint: show that ged(m, n) = n.] (d) Show that ord(a) divides p-1. [Hint: you may use Fermat's Little Theorem from
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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C and d ASAP
![2. Let p be a prime and let a € F,- {0}.
(a) Show that there exists a natural number mEN such that a
that the set {a: kN) is finite.]
=
1. [Hint: first show
(b) The order of a, denoted ord(a), is defined to be the smallest natural number n such
that a" = 1. What are the orders of the elements of F,- {0} for p = 3,5,7.
(e) Show that if a" = 1 for some m € Z then n = ord(a) divides m. [Hint: show that
ged(m, n) = n.]
(d) Show that ord(a) divides p - 1. [Hint: you may use Fermat's Little Theorem from](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad4d6d9a-c2bc-4670-b756-591f350e4f00%2F195d7d3f-d8c2-4543-b428-88b0719df251%2Fkov3kci_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Let p be a prime and let a € F,- {0}.
(a) Show that there exists a natural number mEN such that a
that the set {a: kN) is finite.]
=
1. [Hint: first show
(b) The order of a, denoted ord(a), is defined to be the smallest natural number n such
that a" = 1. What are the orders of the elements of F,- {0} for p = 3,5,7.
(e) Show that if a" = 1 for some m € Z then n = ord(a) divides m. [Hint: show that
ged(m, n) = n.]
(d) Show that ord(a) divides p - 1. [Hint: you may use Fermat's Little Theorem from
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