Axiomatic characterization of fields by the product formula for valuations

On equations in two S-units over function fields of characteristic 0

“…By applying (1) with a= 2 we obtain that ML contains at most finitely many archimedean valuations. Moreover, from (1) . and the above arguments we infer that laolv, ... , lanlvE{O, 1} f~r all but_fi~tel~ many nonarchimedean valuations !·Iv in ML· Together with (5) this implies that lfilv = 1 for all but finitely many V in ML.…”

“…If confusion can not arise, we shall not make a proper distinction between a valuation I ·I., and its index v. If all valuations in Mx are raised to the same power then we obtain another set of valuations satisfying (1) and (2) which is called equivalent to MK. There are fields with inequivalent sets of valuations satisfying (1) and (2).…”

“…Artin and Whaples [1] showed that any field K which has a set of valuations MK with at least one archimedean valuation satisfying the finiteness property and the product formula must be an algebraic number field. Fields with a product formula which are not algebraic number fields, namely those for which all valuations are non-archimedean, are called F-jields.…”

“…Before we are able to state the results, we have to mention some facts about these fields. A field K is called a field with a product formula if it has characteristic 0 and if it is endowed with a set of pairwise non-equivalent (multiplicative) valuations MK = {I ·Iv} on K such that for all ex in K*: (1) (1) If K is a field then we denote by K* the set of non-zero -elements of K. (1) lcxl 11 = 1 for all but finitely many v, the .finiteness property; (2) n l1Xlv = 1, the product formula.…”

On the Rings of Valuation Vectors

“…If such a function v(U, V) is given and if W is any open, linearly compact subspace contained in both U and V, it follows, from (1), (2), that…”