# Galois Groups of Extensions by Roots of Unity

1. Apr 24, 2012

### lugita15

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.

2. Apr 24, 2012

### mathwonk

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
3. Apr 24, 2012

### lugita15

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
4. Apr 24, 2012

### DonAntonio

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

5. Apr 24, 2012

### lugita15

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?

6. Apr 24, 2012

### lugita15

I'm also curious to know, under what circumstances is the unit group of a finite ring cyclic, in general?

7. Apr 27, 2012

### mathwonk

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
8. Apr 27, 2012

### lugita15

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