Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Galois Groups of Extensions by Roots of Unity

  1. Apr 24, 2012 #1
    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.
     
  2. jcsd
  3. Apr 24, 2012 #2

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper

    read these notes starting from page 46.

    http://www.math.uga.edu/%7Eroy/843-2.pdf [Broken]
     
    Last edited by a moderator: May 5, 2017
  4. Apr 24, 2012 #3
    Thanks mathwonk! So now the question becomes, for what n is the unit group of Zn cyclic?
     
    Last edited by a moderator: May 5, 2017
  5. Apr 24, 2012 #4


    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
     
  6. Apr 24, 2012 #5
    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?
     
  7. Apr 24, 2012 #6
    I'm also curious to know, under what circumstances is the unit group of a finite ring cyclic, in general?
     
  8. Apr 27, 2012 #7

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper

    read the following notes, pages 48-50.

    http://www.math.uga.edu/%7Eroy/844-2.pdf [Broken]
     
    Last edited by a moderator: May 5, 2017
  9. Apr 27, 2012 #8
    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?
     
    Last edited by a moderator: May 5, 2017
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Galois Groups of Extensions by Roots of Unity
  1. Galois extension (Replies: 1)

  2. Galoi group (Replies: 4)

Loading...