Editing Field (mathematics) (section)

== Invariants of fields ==
Basic invariants of a field {{math|''F''}} include the characteristic and the [[transcendence degree]] of {{math|''F''}} over its prime field. The latter is defined as the maximal number of elements in {{math|''F''}} that are algebraically independent over the prime field. Two algebraically closed fields {{math|''E''}} and {{math|''F''}} are isomorphic precisely if these two data agree.<ref>{{harvp|Gouvêa|2012|loc=Theorem 6.4.8}}</ref> This implies that any two [[uncountable]] algebraically closed fields of the same [[cardinality]] and the same characteristic are isomorphic. For example, {{math|{{overline|'''Q'''}}<sub>''p''</sub>, '''C'''<sub>''p''</sub>}} and {{math|'''C'''}} are isomorphic (but ''not'' isomorphic as topological fields).

=== 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>

=== Absolute Galois group ===
For fields that are not algebraically closed (or not separably closed), the [[absolute Galois group]] {{math|Gal(''F'')}} is fundamentally important: extending the case of finite Galois extensions outlined above, this group governs ''all'' finite separable extensions of {{math|''F''}}. By elementary means, the group {{math|Gal('''F'''<sub>''q''</sub>)}} can be shown to be the [[Prüfer group]], the [[profinite completion]] of {{math|'''Z'''}}. This statement subsumes the fact that the only algebraic extensions of {{math|Gal('''F'''<sub>''q''</sub>)}} are the fields {{math|Gal('''F'''<sub>''q''<sup>''n''</sup></sub>)}} for {{math|''n'' > 0}}, and that the Galois groups of these finite extensions are given by
: {{math|1=Gal('''F'''<sub>''q''<sup>''n''</sup></sub> / '''F'''<sub>''q''</sub>) = '''Z'''/''n'''''Z'''}}.
A description in terms of generators and relations is also known for the Galois groups of {{math|''p''}}-adic number fields (finite extensions of {{math|'''Q'''<sub>''p''</sub>}}).<ref>{{harvp|Jannsen|Wingberg|1982}}</ref>

[[Galois representation|Representations of Galois groups]] and of related groups such as the [[Weil group]] are fundamental in many branches of arithmetic, such as the [[Langlands program]]. The cohomological study of such representations is done using [[Galois cohomology]].<ref>{{harvp|Serre|2002}}</ref> For example, the [[Brauer group]], which is classically defined as the group of [[central simple algebra|central simple {{math|''F''}}-algebras]], can be reinterpreted as a Galois cohomology group, namely
: {{math|1=Br(''F'') = H<sup>2</sup>(''F'', '''G'''<sub>m</sub>)}}.

=== K-theory ===
[[Milnor K-theory]] is defined as
: <math>K_n^M(F) = F^\times \otimes \cdots \otimes F^\times / \left\langle x \otimes (1-x) \mid x \in F \smallsetminus \{0, 1\} \right\rangle.</math>
The [[norm residue isomorphism theorem]], proved around 2000 by [[Vladimir Voevodsky]], relates this to Galois cohomology by means of an isomorphism
: <math>K_n^M(F) / p = H^n(F, \mu_l^{\otimes n}).</math>
[[Algebraic K-theory]] is related to the group of [[invertible matrix|invertible matrices]] with coefficients the given field. For example, the process of taking the [[determinant (mathematics)|determinant]] of an invertible matrix leads to an isomorphism {{math|1=''K''<sub>1</sub>(''F'') = ''F''<sup>×</sup>}}. [[Matsumoto's theorem (K-theory)|Matsumoto's theorem]] shows that {{math|''K''<sub>2</sub>(''F'')}} agrees with {{math|''K''<sub>2</sub><sup>''M''</sup>(''F'')}}. In higher degrees, K-theory diverges from Milnor K-theory and remains hard to compute in general.