This is very embarrassing thing to ask but what is a definition of global field? Every text or internet sources says either a number field or function field over finite field. (Yes, I understand there is nothing wrong with this but I would like to know the 'axiomatic' definition)
Thanks,
I glanced at the paper, and here's a summary. There are two axioms imposed on a field $k$ which are necessary and sufficient (these are theorems in the paper) for the field to be global (i.e. a number field or a finitely generated extension of a finite field of transcendence degree $1$):
Axiom (1): There is a set $\mathfrak{M}$ of places of $k$ (equivalence classes of absolute values) and absolute values $\vert \cdot\vert_\mathfrak{p}\in\mathfrak{p}$ for each $\mathfrak{p}\in\mathfrak{M}$ such that for each $\alpha\neq 0$ in $k$, $\vert \alpha\vert_\mathfrak{p}=1$ for all but finitely many $\mathfrak{p}$, and we have the product formula
$$\prod_{\mathfrak{p}\in\mathfrak{M}}\vert \alpha\vert_\mathfrak{p}=1.$$
Axiom (2) The set $\mathfrak{M}$ contains at least one place $\mathfrak{p}$ which is either discrete with finite residue field or Archimedean.
Theorem 2 of the paper says that algebraic number fields and algebraic function fields satisfy the two axioms, with the every member of the set set of places in each case satisfying Axiom (2) (not just the one place required by Axiom (2)). Theorem 3 says that a field satisfying both axioms is a global field (the converse of Theorem 2).
I think it's probably safe to say that this axiomatic characterization, while nice, is not what most number theorists adopt as the definition of a global field.
