Galois Groups of Extensions by Roots of Unity

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Discussion Overview

The discussion revolves around the Galois groups of field extensions of the form Q(u), where Q is the field of rational numbers and u is the principal nth root of unity. Participants explore under what conditions the Galois group of Q(u) over Q is cyclic, particularly focusing on the values of n that satisfy this condition. The conversation also touches on related topics such as the cyclic nature of unit groups in modular arithmetic and finite rings.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that the Galois group of Q(u) over Q is cyclic when n is prime or twice an odd prime, but they seek to identify other values of n that may also satisfy this condition.
  • It is noted that the Galois group Gal(Q(u)/Q) is isomorphic to the unit group (Z/nZ)*, leading to the question of when these unit groups are cyclic.
  • DonAntonio asserts that the unit group of Z/nZ is cyclic if n is a prime, a power of an odd prime, or twice a prime, suggesting that these are the only cases where the Galois group is cyclic.
  • A participant inquires about the proof for the cyclic nature of the unit group of Z/nZ when n is a power of an odd prime, indicating a desire for further clarification on this point.
  • Another participant expresses curiosity about the general conditions under which the unit group of a finite ring is cyclic, expanding the discussion beyond just the case of Z/nZ.

Areas of Agreement / Disagreement

Participants generally agree on some specific cases where the Galois group is cyclic, but there is no consensus on the complete characterization of all such n. The discussion remains unresolved regarding the broader implications for unit groups in finite rings.

Contextual Notes

Participants reference external notes and resources to support their claims, indicating that the discussion may depend on specific definitions and theorems related to group theory and field extensions.

lugita15
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Consider field extensions of the form Q(u) where Q is the field of rational numbers and u=e^{\frac{2\pi i}{n}}, 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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read these notes starting from page 46.

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

http://www.math.uga.edu/%7Eroy/843-2.pdf
Thanks mathwonk! So now the question becomes, for what n is the unit group of Zn cyclic?
 
Last edited by a moderator:
lugita15 said:
Consider field extensions of the form Q(u) where Q is the field of rational numbers and u=e^{\frac{2\pi i}{n}}, 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 Gal\left(\mathbb Q(u)/\mathbb Q\right) \cong \left(\mathbb Z/n\mathbb Z\right)^{*} , 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 \phi(n)

DonAntonio
 
DonAntonio said:
As Gal\left(\mathbb Q(u)/\mathbb Q\right) \cong \left(\mathbb Z/n\mathbb Z\right)^{*} , 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 \phi(n)

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

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

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