Let n = pq where p and q are two distinct primes and consider Zn. Further, let [a] E Zn be an invertible element of order w. Recall that Carmichael's A-function, A(n), is defined to be the smallest positive integer m such that [æ]" = [1] for every invertible element [x] E Zn. Prove that w/A(n).
Let n = pq where p and q are two distinct primes and consider Zn. Further, let [a] E Zn be an invertible element of order w. Recall that Carmichael's A-function, A(n), is defined to be the smallest positive integer m such that [æ]" = [1] for every invertible element [x] E Zn. Prove that w/A(n).
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.6: Algebraic Extensions Of A Field
Problem 7E
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Question
![Let n = pq where p and q are two distinct primes and consider Zn.
Further, let a] E Zn be an invertible element of order w.
Recall that Carmichael's -function, X(n), is defined to be the smallest positive
integer m such that
[æ]" = [1]
for every invertible element [x] E Zn. Prove that
w/A(n).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe8260224-635c-426e-a960-28f2cf28de1f%2Fbdb6e91d-464e-48c8-98e5-83d2924de551%2Fxjirump_processed.png&w=3840&q=75)
Transcribed Image Text:Let n = pq where p and q are two distinct primes and consider Zn.
Further, let a] E Zn be an invertible element of order w.
Recall that Carmichael's -function, X(n), is defined to be the smallest positive
integer m such that
[æ]" = [1]
for every invertible element [x] E Zn. Prove that
w/A(n).
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