Franklin's Notes


P-adic metric

The p-adic absolute value on $\mathbb{Q}$ $|\cdot|_p$ is induced by the p-adic valuation , and is defined as where $|0|_p$ is taken to be $0$ by definition. This endows the rational numbers with a metric defined by turning it into a metric space, so that the p-adic numbers can be defined as the completion of $\mathbb{Q}$ with respect to $|\cdot|_p$.

It follows form the properties of the p-adic valuation that this metric satisfies the ultrametric inequality This has a couple of strange implications on the topology of the p-adic numbers $\mathbb{Q}_p$. In particular:

abstract-algebra

metric-space

analysis

topology

p-adic-analysis

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