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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

  1. Lang, Serge: "Algebra - Graduate Texts in Mathematics", Springer, 2002, 3rd Edition