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FIELDS ADMITTING NONTRIVIAL STRONG ORDERED EULER CHARACTERISTICS ARE QUASIFINITE

The note contains the details of an assertion made in [1] to the effect that fields admitting a nontrivial strong ordered Euler characteristic are quasifinite. In this section we recall the relevant definitions and in the next section we complete the proof. Recall that a field K is quasifinite if K is perfect and its absolute Galois group is isomorphic to the profinite completion of Z. In particular, a finite field is quasifinite. A strong ordered Euler characteristic on the field K is a function χ : Def(K)→ R from the set of definable (in the language of rings) subsets of (any Cartesian power) of K to a partially ordered ring R having image amongst the nonnegative elements of R and satisfying χ(X) = χ(Y ) for X and Y definably isomorphic, χ(X × Y ) = χ(X) · χ(Y ), χ(X ∪ Y ) = χ(X) + χ(Y ) for X ∩ Y = ∅, and χ(E) = c · χ(B) if f : E → B is a definable function and c = χ(f−1{b}) for every b ∈ B. The Euler characteristic is nontrivial if 0 < 1 in R and the image of χ is not just {0}. The main theorem of this note is the following: