Proposition 10

Let $K$ be a field, $f$ an absolute value on $K$ such that $K$ is complete and not discrete with respect to $f$ and $L$ an algebraic extension of $K$. Then $f$ can be extended uniquely to an absolute value $f'$ on $L$ and, if $L$ is of finite degree $n$, then

for all $x \in L$.

The existence of $f'$ follows from Proposition 9 and its uniqueness (over every finite sub-extension of $L$ and therefore over the whole of $L$) from Lemma 2 of $§ 6$, no. 4. Let $f'$ be the unique extension off to the algebraic closure of $K$ and suppose $L$ is of finite degree $n$. We know that $N_{L/K}(x) = \prod_{i=1}^n x_i$, where each $x_i$ is a conjugate of $x$ over $K$ (Algebra, Chapter VIII, $§ 12$, no. 2, Proposition 4). In view of the uniqueness of $f'$, $f'(x_i) = f'(x)$ for all $i$, whence the stated formula.