Question.
Prove that the multiplicative group of any infinite field can never be cyclic .
$\mathbb R$, $\mathbb Q$, $\mathbb C$ are some infinite fields whose multiplicative groups are not cyclic, I know.
I need some lead as to how to begin the proof.
Sorry for the lack of work on my part (I'm clueless) and any help is appreciated.
Best Answer
Ok here is the characteristic 2 case:
Assume $k$ is an infinite field of characteristic $2$ with a cyclic multiplicative group. Note that any element of an algebraic extension of $\mathbb{F}_2$ has finite multiplicative order, so this implies that every element of $k-\{0,1\}$ must be transcendental.
Next let $x$ be a generator for the multiplicative group, which exists as we are assuming it is cyclic. Consider the element $1+x$ of our field. It is nonzero and therefore equal to some power of $x$ since $x$ is a generator. But then $1+x=x^n$ for some $n$, so $x$ is algebraic over $\mathbb{F}_2$, contradicting the above claim.