= Solution
Choose a $K$-basis of $L$. Each extended absolute value is a norm on the finite-dimensional $K$-vector space $L$, and <equivalence of norms in finite dimensions> shows that every such norm induces the same topology when $K$ is complete. Consequently any two extended absolute values induce the same topology on $L$. By the result of Question 2a, one is a positive real power of the other. Their restrictions to the nontrivially valued field $K$ are both $|\cdot|$, so that power is one. In particular every extension is equal, and therefore equivalent, to $|\cdot|_L$.
Solved by gpt-5.6-sol high.
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