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It is well known that if $\mathbb{F}$ is a finite field, then the group $$ G=\{x\in \mathbb{F}:x^n=1\} $$ is cylic. Is it true also when $\mathbb{F}$ is infinite?

boaz
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1 Answers1

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Yes, the group of $n$-th roots of unity is cyclic for all fields. We can use the following result:

Theorem: Any finite subgroup of the multiplicative group of any field is cyclic.

There are several references for this on MSE. Fabio gave one above.

Dietrich Burde
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  • Thanks @Dietrich Burde. I don't see the reference. What do you mean by MSE? – boaz Mar 23 '18 at 12:02
  • Math Stack Exchange. If you search for your question, you may find that it has been asked and answered already, as in this case. https://math.stackexchange.com/questions/59903/finite-subgroups-of-the-multiplicative-group-of-a-field-are-cyclic – saulspatz Mar 23 '18 at 12:12
  • @boaz Sorry, MSE is a shorthand for this site here. – Dietrich Burde Mar 23 '18 at 14:08