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Proposition: Characterization of Dependent Absolute Values
Let $(F,+,\cdot)$ be a field with two absolute values $|\cdot|_1$ and $|\cdot|_2$ defined on it. If $|\cdot|_1$ and $|\cdot|_2$ are dependent, then there exists a positive real number $\lambda > 0$ such that $$|x|_1=|x|_y^\lambda$$
for all $x\in F$ with $x\neq 0.$
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- Lang, Serge: "Algebra - Graduate Texts in Mathematics", Springer, 2002, 3rd Edition