Is $F^\times/N_{L/F}(L^\times)$ non trivial or infinite when $L/F$ is an extension of global fields? Let $L/F$ be an extension of global fields of degree $n\geq 2$, not necessarily Galois.
I thought it was well known that the group $F^\times/N_{L/F}(L^\times)$ is always non trivial, and even infinite.
However, I am not able to find any reference pointing in this direction, and I can't either come up with a proof. I convinced myself that this group is non trivial when $L/F$ is cyclic, but that's it. Hence my questions:
Questions.

*

*Is is true that $F^\times/N_{L/F}(L^\times)$ is always non trivial ?


*Is this group finite? infinite ?
Some evidence suggests that the answer to 1. is yes ( it is true for cyclic extensions, I think, and for extensions such that $F$ is real and complex conjugation is a non trivial automorphism of $L$)
 A: For number fields you can do it with Chebotarev.
If $N_{L/F}(L^\times)$ has finite index in $F^\times$ then $N_{L/F}(\mathcal{P}_L)$ has finite index in $\mathcal{P}_F$ (the group of fractional principal ideals) whence in $\mathcal{I}_F$ (the group of fractional ideals) so that $N_{L/F}(\mathcal{I}_L)$ has finite index in $\mathcal{I}_F$.
Let $e_{p,K}$ be the number of primes of $O_K$ of norm $p$ a prime in $\Bbb{Z}$.

Chebotarev says that there is a positive density of $p$ such that $e_{p,L}=[L:\Bbb{Q}]$ (which implies that $e_{p,F}=[F:\Bbb{Q}]$) so that $$\lim_{s\to 1}\prod_{p,\ \ e_{p,L}=[L:\Bbb{Q}]} \frac{(1-p^{-s})^{e_{p,F}}}{(1-p^{-s})^{e_{p,L}}}=\infty$$ He also says that $\zeta_L(s)/\zeta_F(s)$ whence $\prod_p \frac{(1-p^{-s})^{e_{p,F}}}{(1-p^{-s})^{e_{p,L}}}$ is bounded as $s \to 1$.

This implies that there are infinitely many $p$ such that $e_{p,L}<e_{p,F}$, which implies that there are infinitely many primes $\mathfrak{p}\subset O_F$ that are not a norm  of any prime $\mathfrak{P}\subset O_L$,
which implies that $N_{L/F}(\mathcal{I}_L)$ has infinite index in $\mathcal{I}_F$.
