 Solution

Let $e_1,\dots ,e_d$ be a ${\mathbb{Q}}_p$-basis of $K$, and define the coordinate normFirst an extending absolute value on a field is automatically non-Archimedean. Its values on integers are at most one, and the binomial theorem givesTaking $m$th roots and letting $m\to \infty$ proves the ultrametric inequality.

ConsequentlyThis bound makes $x\mapsto |x|$ continuous for the coordinate topology. The coordinate sphere $S=\{x:\Vert x{\Vert }_0=1\}$ is a closed subset of ${\mathbb{Z}}_p^d$, hence compact. It contains no zero, so the continuous positive function $|x|$ attains a positive minimum $c$ there. Scaling any nonzero vector by a power of $p$ to put it in $S$ provesThis is the compact-sphere proof of finite-dimensional non-Archimedean norm equivalence.

A Cauchy sequence for $|\cdot |$ is therefore Cauchy in each coordinate. The completeness of ${\mathbb{Q}}_p$ gives a coordinate limit, and the upper bound gives convergence to it for $|\cdot |$. Thus every extending absolute value makes $K$ complete.

For uniqueness, if $|\cdot {|}_1$ and $|\cdot {|}_2$ are two extensions, their respective bounds give constants $A,B>0$ withApply this to $x^m$ and use multiplicativity:Letting $m\to \infty$ proves $\boxed{|x{|}_1=|x{|}_2}$ for every $x$. This proves the unique extension of an absolute value to a finite extension without assuming that an arbitrary coordinate norm is multiplicative.