Index of norm group of a global field For a global field $K$ (characteristic $p$ or $0$), is there anything meaningful which could be said about the value $[K^\times:N_{L/K}L^\times]$ where $L$ is a finite extension (possibly of prime degree).
Here is separate question which really was (somehow) the motivation for the question asked above.. Feel free to ignore this one though!
I was trying to figure out why explicit computation of $\#H^1(G(L/K),C_L)=1$ is hard (as far as I know it has not been done) and it led to (possibly false) conclusion that this is due to the fact that if an element $x\in K\subset K_P$ is a norm from $L_\mathfrak{P}$, say $x=N_{L_{\mathfrak{P}}/K_P}(y)$ then $y$ is not necessarily related to a global element(By this I mean that $x$ is not necessarily of the form $N_{L_{\mathfrak{P}}/K_P}(y)+(1-\sigma)(z)$ for some $y\in L$ and $z\in L_\mathfrak{P}$.) Here of course $\mathfrak{P}$ is a place of $L$ dividing place $P$ of $K$. 
So question here would be, whether what I said is actually true. i.e. is it not necessary that an element which its norm is a global element to be (related to) a global element?
 A: For a Galois extension $L/K $ of global fields of degree $n$ and group $G$, the $G$-cohomology of the idèle class group $C_L$ is entirely known : $H^1(G, C_L) = 0$ ,  $H^2(G, C_L)$ is cyclic of degree $n$, with a canonical  generator $u_{L/K}$ called the « fundamental class » ; if these groups, as you say, are so hard to determine, that’s because they give all the other Tate comology (which I’ll write without the usual hat) : for any $r\in \mathbf Z$, the cup-product with $u_{L/K}$ gives an isomorphism $H^r(G, \mathbf Z) \cong H^{r+2}(G, C_L)$ , where the LHS can be considered as explicitly known if we are given enough information on $G$. This is the formalism of « class formations » (see last chapter of Artin-Tate), which applies also to to local extensions, replacing $C_L$ by the local multiplicative group $L_v^*$ . Since the idèle group $J_L$ is the restricted product of the local multiplicative groups, the $G$-cohomology of $J_L$ is just the direct sum of the local cohomologies of the local multiplicative groups, so we can say that it’s also « explicitly known ». 
Let us apply this to the problem of computing the « normic quotient » $K^* /N(L^*)$ by local-global methods. The exact sequence $1 --> L^* -->J_L -->C_L -->1$ gives rise to maps $g_{r-1} : H^{r-1}(G, J_L)--> H^{r-1}(G, C_L)$ and $f_r : H^r(G, L^*)-->H^{r}(G, J_L)$ such that ker$f_r$ = coker$g_{r-1}$ . For any $r$, ker $f_r$ is dual to the kernel of $h : H^{3-r}(G, \mathbf Z)$ --> direct product of $H^{3-r}(G, \mathbf Z)$ (see p. 198 of Tate’s chapter VII in Cassels-Fröhlich’s book). In the special case $r = 0$, ker$f_0$ = the quotient {elements of $K^*$ which are local norms everywhere} / {elements which are global norms} = coker$g_{1}$, which gives in principle the norm group $N(L^*)$ if we know $G$ and ramification- decomposition in $L/K$. As for coker$f_0$, because of Hilbert 90, it's $\cong H^0(G, C_L) \cong H^{-2}(G, \mathbf Z)$, which gives in principle the normic quotient $K^* /N(L^*)$. For instance, if $G$ is cyclic, then $H^3(G, \mathbf Z) = Hom(G, \mathbf Z) = 0$ and one recovers Hasse’s norm principle and coker$f_0$ is cyclic of order $n$ ; if $G$ is not cyclic, this principle does not always hold (see example in op. cit., exercise 5).
