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It's known that local fields can be defined as a non-discrete, Hausdorff (equivalently non-indiscrete), locally compact, topological field, which is the same as non-trivial (i.e. neither discrete nor indiscrete) locally compact topological field.

In particular, local fields can be defined without mentioning absolute values. For me, absolute values involve the non-natural group $\mathbb{R}$ as the codomain, making those valued fields bounding to $\mathbb{R}$ in some way.

We know global fields can be defined using local fields and absolute valuations. Is it possible to define global fields as certain type of topological fields without mentioning any absolute value?

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edit: "without mentioning any absolute value" means that the axiomatic definition of global field doesn't involve absolute values.

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    $\begingroup$ Why not just define global fields as the fields of fractions of Dedekind domains of finite type over $\mathbb Z$? $\endgroup$
    – Will Sawin
    Commented Jun 1, 2023 at 20:27
  • $\begingroup$ @WillSawin That sounds like a cool definition! Is there a reference using that definition or proving this equivalence? And is there a similar one for local fields? $\endgroup$
    – Z Wu
    Commented Jun 1, 2023 at 20:41
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    $\begingroup$ Global fields do not come with a preferred topology: I believe no definition of a global field will mention any absolute value. $\endgroup$
    – Chris Wuthrich
    Commented Jun 1, 2023 at 20:41
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    $\begingroup$ What definition of global field are you familiar with? $\endgroup$
    – Will Sawin
    Commented Jun 1, 2023 at 20:44
  • $\begingroup$ @WillSawin the axiomatic definition by Artin-Whaples, and the usual definition that list all the global fields. $\endgroup$
    – Z Wu
    Commented Jun 1, 2023 at 20:53

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