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Commutative finite ring and the EulerLagrange Theorem 
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#1
Mar2612, 10:25 AM

P: 6

1. The problem statement, all variables and given/known data
We are given the ring [itex]Z[/itex]/1026[itex]Z[/itex] with the ordinary addition and multiplication operations. We define G as the group of units of [itex]Z[/itex]/1026[itex]Z[/itex]. We are to show that g[itex]^{18}[/itex]=1. 2. Relevant equations The Eulerphi (totient) function, here denoted [itex]\varphi[/itex](n) 3. The attempt at a solution I have verified that G is indeed a group and concluded that G contains all elements of [itex]Z[/itex]/1026[itex]Z[/itex] coprime to 1026. I also know from the EulerLagrange theorem that since every g[itex]\in[/itex]G is coprime to 1026, g[itex]^{\varphi(1026)}[/itex]=1 (mod 1026). [itex]\varphi(1026)[/itex]=18*18*2=648 [itex]\Rightarrow[/itex] g[itex]^{18*18*2}[/itex]=1 (mod 1026) (g[itex]^{18}[/itex])[itex]^{18*2}[/itex]=1 (mod 1026) (g[itex]^{18}[/itex])[itex]^{18}[/itex])(g[itex]^{18}[/itex])[itex]^{18}[/itex])=1 (mod 1026) So the element (g[itex]^{18}[/itex])[itex]^{18}[/itex]) is necessarily the identity or of order 2. A simple check shows that there are no integers h[itex]\in[/itex]G between 2 and 1025 such that h=(g[itex]^{18}[/itex])[itex]^{18}[/itex])=[itex]\sqrt{1026n+1}[/itex], n some positive integer. Thus (g[itex]^{18}[/itex])[itex]^{18}[/itex])=1 (mod 1026). (Is the above reasoning correct?) And here begins my trouble. I wish somehow to show that the greatest order of any element in G is 18 and that any other orders are composed of prime factors of 18. I suppose that the fundamental theorem of finitely generated abelian groups is of limited use here, since there are som many possible combinations of prime factors. I figure that the totient function will be of some aid, but I can't find a reason as for why there can't for instance be any elements in G of order 12 or 27. What makes 18 (or 9,6,3,2,1) so special? Could someone please give me a tip? Thanks! 


#2
Mar2812, 09:30 PM

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##\phi(1026)=18^2## and not ##18^2 \cdot 2##.



#3
Mar2912, 09:54 AM

P: 6

Thank you morphism.
I did notice myself yesterday that phi(1026)=18^2. However, I could not deduce why there are no elements in G of order greater than 18. The homework is already due and I will have it corrected and commented soon. But if someone still feels like coming up with suggestions you are very welcome. I simply can't get why there are no elements for which g^18[itex]\neq[/itex]1. For instance, can't there be any elements of order 12 or 27? 


#4
Mar2912, 06:58 PM

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Commutative finite ring and the EulerLagrange Theorem
Probably the easiest way to approach this is to note that we have the ring isomorphism $$ \mathbb Z / 1026 \mathbb Z \cong \mathbb Z/2 \mathbb Z \times \mathbb Z/ 3^3 \mathbb Z \times \mathbb Z/19 \mathbb Z $$ (this comes from the chinese remainder theorem) and hence the isomorphisms
$$ (\mathbb Z / 1026 \mathbb Z)^\times \cong (\mathbb Z/2 \mathbb Z)^\times \times (\mathbb Z/ 3^3 \mathbb Z)^\times \times (\mathbb Z/19 \mathbb Z)^\times \cong \mathbb Z/ (3^3  3^2) \mathbb Z \times \mathbb Z/(191) \mathbb Z \cong \mathbb Z/18 \mathbb Z \times \mathbb Z/18 \mathbb Z $$ of the unit groups. 


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