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Extending 2-adic valuation to real numbers So i know $\mathbb{R}$ has $p$-adic valuation

But could we have absolute value $|\cdot|$ to $p$-adic number?

flower
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    If this is your question then yes $\Bbb{C}$ contains all the transcendental extensions of $\Bbb{Q}$ with a not too large cardinality, in particular it contains a copy of $\Bbb{Q}_p$, but the embedding $\Bbb{Q_p\to C}$ is an abstract nonsense requiring the axiom of choice. Conversely the $|.|_p$ completion of $\overline{\Bbb{Q}}_p$ contains a copy of $\Bbb{C}$. – reuns Jan 01 '20 at 12:32
  • Oh.. absolute value mean usual real absolute value sorry my question is $\mathbb{Q}$ has very usual absolute value and it is still work to $\Bbb{Q_p}$ – flower Jan 01 '20 at 12:53
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    The embedding defines the absolute value... For example $|f(x)|_\infty = |f(\pi)|$ is an absolute value on $\Bbb{Q}(x)$ – reuns Jan 01 '20 at 13:01

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