Timeline for $n$-th root of character on local field

3 events

when toggle format	what		by	license	comment
Jan 5 at 15:53	comment	added	Windi		Thanks! This makes sense.
Jan 4 at 13:49	comment	added	sdr		Suppose $A \subset B$ are abelian groups, and $\Gamma$ is a divisible group. Then it's standard that any homomorphism $A \rightarrow \Gamma$ extends to $B$ -- this is Baer's criterion for $\mathbb{Z}$. Now take $B = F^{\times}$, $A$ the subgroup of $n$th powers, and $\Gamma = \mathbb{C}^{\times}$ to get the claim you want. If $\chi$ is trivial on $\mu_n(F) :=$ the $n$th roots of unity of $F^{\times}$ (without assumptions -- there may not exist any), then it factors through $F^{\times}/\mu_n(F) = A$, which by the above extends to $B$, which was your question.
Jan 4 at 9:13	history	asked	Windi	CC BY-SA 4.0