Elementary equivalence versus Isomorphism

1) The isomorphy type of a finite field K is given by its cardinality |K|, i.e., if K and L are such fields, then K ∼= L iff |K| = |L|. 1) The isomorphy type of an algebraically closed field K is determined by two invariants: (i) Absolute transcendence degree td(K), (ii) The characteristic p = char(K) ≥ 0. In other words, if K and L are algebraically closed fields, then K ∼= L iff td(K) = td(L) and char(K) = char(L). Nevertheless, if we want to classify fields K up to isomorphism even in an “only a little bit” more general context, then we run into very serious difficulties... A typical example here is the attempt to give the isomorphy types of real closed fields (it seems first tried by Artin and Schreier). A real closed field K is “as close as possible” to its algebraic closure K, as [K : K] = 2, and K = K[ √ −1]. These fields were introduced by Artin in his famous proof of Hilbert’s 17th Problem. Roughly speaking, the real closed fields are the fields having exactly the same algebraic properties as the reals ' . One knows quite a lot about real closed fields, see e.g. Prestel–Roquette [P–R], but for specialists it is clear that the problem of describing the isomorphism types of all such fields is hopeless... And to move into more modern times, the same is true for the p-adically closed fields (which roughly speaking, are the fields having exactly the same algebraic properties as the p-adics