Editing Field (mathematics) (section)

=== Model theory of fields ===
In [[model theory]], a branch of [[mathematical logic]], two fields {{math|''E''}} and {{math|''F''}} are called [[elementarily equivalent]] if every mathematical statement that is true for {{math|''E''}} is also true for {{math|''F''}} and conversely. The mathematical statements in question are required to be [[first-order logic|first-order]] sentences (involving {{math|0}}, {{math|1}}, the addition and multiplication). A typical example, for {{math|''n'' > 0}}, {{math|''n''}} an integer, is
: {{math|''φ''(''E'')}} = "any polynomial of degree {{math|''n''}} in {{math|''E''}} has a zero in {{math|''E''}}"
The set of such formulas for all {{math|''n''}} expresses that  {{math|''E''}} is algebraically closed.
The [[Lefschetz principle]] states that {{math|'''C'''}} is elementarily equivalent to any algebraically closed field {{math|''F''}} of characteristic zero. Moreover, any fixed statement {{math|''φ''}} holds in {{math|'''C'''}} if and only if it holds in any algebraically closed field of sufficiently high characteristic.<ref>{{harvp|Marker|Messmer|Pillay|2006|loc=Corollary 1.2}}</ref>

If {{math|''U''}} is an [[ultrafilter]] on a set {{math|''I''}}, and {{math|''F''<sub>''i''</sub>}} is a field for every {{math|''i''}} in {{math|''I''}}, the [[ultraproduct]] of the {{math|''F''<sub>''i''</sub>}} with respect to {{math|''U''}} is a field.<ref>{{harvp|Schoutens|2002|loc=§2}}</ref> It is denoted by
: {{math|ulim<sub>''i''→∞</sub> ''F''<sub>''i''</sub>}},
since it behaves in several ways as a limit of the fields {{math|''F''<sub>''i''</sub>}}: [[Łoś's theorem]] states that any first order statement that holds for all but finitely many {{math|''F''<sub>''i''</sub>}}, also holds for the ultraproduct. Applied to the above sentence {{math|φ}}, this shows that there is an isomorphism{{efn|Both {{math|'''C'''}} and {{math|ulim<sub>''p''</sub> {{overline|'''F'''}}<sub>''p''</sub>}} are algebraically closed by Łoś's theorem. For the same reason, they both have characteristic zero. Finally, they are both uncountable, so that they are isomorphic.}}
: <math>\operatorname{ulim}_{p \to \infty} \overline \mathbf F_p \cong \mathbf C.</math>
The Ax–Kochen theorem mentioned above also follows from this and an isomorphism of the ultraproducts (in both cases over all primes {{math|''p''}})
: {{math|ulim<sub>''p''</sub> '''Q'''<sub>''p''</sub> ≅ ulim<sub>''p''</sub> '''F'''<sub>''p''</sub>((''t''))}}.
In addition, model theory also studies the logical properties of various other types of fields, such as [[real closed field]]s or [[exponential field]]s (which are equipped with an exponential function {{math|exp : ''F'' → ''F''<sup>×</sup>}}).<ref>{{harvp|Kuhlmann|2000}}</ref>