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Proposition: Characterization of Non-Archimedean Absolute Values

  1. If $|\cdot|$ is a non-archimedean absolute value of a field $(F,+,\cdot)$, then $|n|\le 1$ for all multiples of $1\in F$, i.e. $$n=\underbrace{1+\ldots+1}_{n\text{ times}},\quad 1\in F.$$
  2. If $|\cdot|$ is a non-archimedean absolute value of a field $(F,+,\cdot)$, then $|n|\le 1$ for all multiples of $1\in F$, i.e. $$n=\underbrace{1+\ldots+1}_{n\text{ times}},\quad 1\in F.$$

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