Galois Groups of Extensions by Roots of Unity

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Consider field extensions of the form Q(u) where Q is the field of rational numbers and [itex]u=e^{\frac{2\pi i}{n}}[/itex], the principal nth root of unity. For what values of n is the Galois group of Q(u) over Q cyclic? It seems to at least hold when n is prime or twice an odd prime, but what else?

Any help would be greatly appreciated.

Thank You in Advance.
 

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  • #2
mathwonk
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read these notes starting from page 46.

http://www.math.uga.edu/%7Eroy/843-2.pdf [Broken]
 
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read these notes starting from page 46.

http://www.math.uga.edu/%7Eroy/843-2.pdf [Broken]
Thanks mathwonk! So now the question becomes, for what n is the unit group of Zn cyclic?
 
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  • #4
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Consider field extensions of the form Q(u) where Q is the field of rational numbers and [itex]u=e^{\frac{2\pi i}{n}}[/itex], the principal nth root of unity. For what values of n is the Galois group of Q(u) over Q cyclic? It seems to at least hold when n is prime or twice an odd prime, but what else?

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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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
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?
 
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I'm also curious to know, under what circumstances is the unit group of a finite ring cyclic, in general?
 
  • #7
mathwonk
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read the following notes, pages 48-50.

http://www.math.uga.edu/%7Eroy/844-2.pdf [Broken]
 
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read the following notes, pages 48-50.

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