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Definition: Dependent and Independent Absolute Values
Let $(F,+,\cdot)$ be a field with two absolute values $\cdot_1$ and $\cdot_2$ defined on it. $\cdot_1$ and $\cdot_2$ are called dependent, if and only if $$x_1 < 1\Longleftrightarrow x_2 < 1$$ for all $x\in F.$ Otherwise, we call them independent.
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Proofs: 1
Propositions: 2
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References
Bibliography
 Lang, Serge: "Algebra  Graduate Texts in Mathematics", Springer, 2002, 3rd Edition