Answered: If a is an element of order n and gcd…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.3: Zeros Of Polynomials

Problem 54E

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If a is an element of order n and gcd (p, n) =1.Then show that a ^p is also of order n

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$a b e t h e e l e m e n t o f G a n d o r d e r o f a n e l e m e n t a i s n \Rightarrow a^{n} = e a n d i f a^{k} = e f o r s o m e k \in Z t h e n n | k . . . . (i) a n d g c d (p, n) = 1 n o w l e t o r d e r (a^{d}) = m \Rightarrow {(a^{d})}^{m} = e a n d i f {(a^{d})}^{k} = e f o r s o m e k \in Z, t h e n m | k . . . (i i) n o w a^{d m} = e f r o m (i i) t h e r e f o r e n | d m f r o m (i) b u t g c d (d, n) = 1, t h e r e f o r e n | m . . . . . . (i i i) a l s o f r o m (i) w e h a v e a^{n} = e t h e r e f o r e a^{n d} = e^{d} \Rightarrow {(a^{d})}^{n} = e t h e r e f o r e m | n . . . . . . (i v) f r o m (i i) f r o m (i i i) a n d (i v), w e g e t m = n t h e r e f o r e o r d e r (a^{d}) = m = n$

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