structural:summarization
Model theory of fields

In model theory, a branch of mathematical logic, two fields E and F are called elementarily equivalent if every mathematical statement that is true for E is also true for F and conversely. The mathematical statements in question are required to be first-order sentences (involving 0, 1, the addition and multiplication). A typical example, for n > 0, n an integer, is

The set of such formulas for all n expresses that E is algebraically closed.The Lefschetz principle states that C is elementarily equivalent to any algebraically closed field F of characteristic zero. Moreover, any fixed statement φ holds in C if and only if it holds in any algebraically closed field of sufficiently high characteristic.

If U is an ultrafilter on a set I, and F is a field for every i in I, the ultraproduct of the F with respect to U is a field. It is denoted by

since it behaves in several ways as a limit of the fields F: Łoś's theorem states that any first order statement that holds for all but finitely many F, also holds for the ultraproduct. Applied to the above sentence φ, this shows that there is an isomorphism

The Ax–Kochen theorem mentioned above also follows from this and an isomorphism of the ultraproducts (in both cases over all primes p)

In addition, model theory also studies the logical properties of various other types of fields, such as real closed fields or exponential fields (which are equipped with an exponential function exp : F → F).