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Any help would be greatly appreciated.

Thank You in Advance.

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- Thread starter lugita15
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- #1

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Any help would be greatly appreciated.

Thank You in Advance.

- #2

mathwonk

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read these notes starting from page 46.

http://www.math.uga.edu/%7Eroy/843-2.pdf [Broken]

http://www.math.uga.edu/%7Eroy/843-2.pdf [Broken]

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- #3

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Thanks mathwonk! So now the question becomes, for what n is the unit group of Zn cyclic?read these notes starting from page 46.

http://www.math.uga.edu/%7Eroy/843-2.pdf [Broken]

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- #4

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Any help would be greatly appreciated.

Thank You in Advance.

As [itex]Gal\left(\mathbb Q(u)/\mathbb Q\right) \cong \left(\mathbb Z/n\mathbb Z\right)^{*}[/itex] , the question is the same as asking what multiplicative

groups of units modulo n are cyclic...

They are precisely when n is: a prime, a power of AN ODD prime and twice a prime. you can check this in most decent group theory books, and

any of these groups is of order [itex]\phi(n)[/itex]

DonAntonio

- #5

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DonAntonio, where would I find the proof that unit group of Zn is cyclic if n is the power of an odd prime, and that the values of n you mentioned are the only ones that work?As [itex]Gal\left(\mathbb Q(u)/\mathbb Q\right) \cong \left(\mathbb Z/n\mathbb Z\right)^{*}[/itex] , the question is the same as asking what multiplicative

groups of units modulo n are cyclic...

They are precisely when n is: a prime, a power of AN ODD prime and twice a prime. you can check this in most decent group theory books, and

any of these groups is of order [itex]\phi(n)[/itex]

DonAntonio

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- #7

mathwonk

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read the following notes, pages 48-50.

http://www.math.uga.edu/%7Eroy/844-2.pdf [Broken]

http://www.math.uga.edu/%7Eroy/844-2.pdf [Broken]

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- #8

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Thanks, I had forgotten that you can apply the structure theorem for abelian groups, since Zn* is obviously abelian. Do you have any thoughts on my other question, namely what are the circumstances in general for a unit group of a finite ring to be cyclic?read the following notes, pages 48-50.

http://www.math.uga.edu/%7Eroy/844-2.pdf [Broken]

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