Multiplicative group of an infinite field is not cyclic 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.

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.
