Theorem 6.1 

(1) ||-||_w1, ... , ||_||_wn are exactly all extensions of ||-||_w to L.

(2) Sum_{i=1}^m[ L_wi : K_v ] = [ L : K ].

Corollary 6.2

For any w|v, we have a map 

Hom_{Kv}(Lw, Kv^bar) ---> Hom_K(L, K^bar)

and the following map is a bijection:

U_{w|v} Hom_{Kv}(Lw, Kv^bar) ---> Hom_K(L, K^bar).

Corollary 6.3

N_{L/K}(x)=Prod_{w|v} N_{Lw/Kv}(x).

Tr_{L/K}(x)=Sum_{w|v} Tr_{Lw/Kv}(x).

Corollary 6.4

For any x in L, ||N_{L/K}(x)||_v=Prod_{w|v} ||x||_w^[Lw : Kv].