Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top


Let F be a field of characteristic $0$ such that $|F:\mathbb Q|=2$, and let U be a finite subgroup of F*, the multiplicative group of F. Show that $|U|$ is 1, 2, 3, 4 or 6.

Attempt at solution:

I know all finite subgroups of the multiplicative group of a field are cyclic. I am trying to consider finite subgroups of $\mathbb Q$* (which I think are just {1}, {1,-1}) and then multiplying by the algebraic element which extends $\mathbb Q$ to F, but I can't quite get a solution.

share|cite|improve this question
Hint: what are the cyclic subgroups of $\Bbb{C}^*$? Which are in two-dimensional extensions of $\Bbb{Q}$? – Chris Eagle Feb 8 '13 at 14:59
There are cyclic subgroups of $\mathbb C$* for any integer n. This can be seen by taking the generator to be the $n^{th}$ root of unity. – FBI Feb 8 '13 at 15:01
Right, and what's the degree over $\Bbb{Q}$ of a primitive $n$th root of $1$? – Chris Eagle Feb 8 '13 at 15:02
The number of co-prime integers with n which are less than or equal to n. So we want values of n so the degree is 2 or less, which are precisely 1, 2, 3, 4, 6. Thanks Chris – FBI Feb 8 '13 at 15:08
up vote 1 down vote accepted

If the order of the group is $n$, then the generator is a root of the $n$th cyclotomic polynomial. The degree of this polynomial is $\phi(n)$ and must be $\le 2$. What can you say about $\phi(n)$ if $p|n$ for some prime $p>3$? And what is $\phi(2^a3^b)$ with $a,b\ge 0$?

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.