Field arithmetic


In mathematics, field arithmetic is a subject that studies the interrelations between arithmetic properties of a and its absolute Galois group.
It is an interdisciplinary subject as it uses tools from algebraic number theory, arithmetic geometry, algebraic geometry, model theory, the theory of finite groups and of profinite groups.

Fields with finite absolute Galois groups

Let K be a field and let G = Gal be its absolute Galois group. If K is algebraically closed, then G = 1. If K = R is the real numbers, then
Here C is the field of complex numbers and Z is the ring of integer numbers.
A theorem of Artin and Schreier asserts that these are all the possibilities for finite absolute Galois groups.
Artin–Schreier theorem. Let K be a field whose absolute Galois group G is finite. Then either K is separably closed and G is trivial or K is real closed and G = Z/2Z.

Fields that are defined by their absolute Galois groups

Some profinite groups occur as the absolute Galois group of non-isomorphic fields. A first example for this is
This group is isomorphic to the absolute Galois group of an arbitrary finite field. Also the absolute Galois group of the field of formal Laurent series C) over the complex numbers is isomorphic to that group.
To get another example, we bring below two non-isomorphic fields whose absolute Galois groups are free.
  • Let C be an algebraically closed field and x a variable. Then Gal is free of rank equal to the cardinality of C.
  • The absolute Galois group Gal is compact, and hence equipped with a normalized Haar measure. For a Galois automorphism s let Ns be the maximal Galois extension of Q that s fixes. Then with probability 1 the absolute Galois group Gal is free of countable rank.
In contrast to the above examples, if the fields in question are finitely generated over Q, Florian Pop proves that an isomorphism of the absolute Galois groups yields an isomorphism of the fields:
Theorem. Let K, L be finitely generated fields over Q and let a: Gal → Gal be an isomorphism. Then there exists a unique isomorphism of the algebraic closures, b: Kalg → Lalg, that induces a.
This generalizes an earlier work of Jürgen Neukirch and Koji Uchida on number fields.

Pseudo algebraically closed fields

A pseudo [algebraically closed field] K is a field satisfying the following geometric property. Each absolutely irreducible algebraic variety V defined over K has a K-rational point.
Over PAC fields there is a firm link between arithmetic properties of the field and group theoretic properties of its absolute Galois group. A nice theorem in this spirit connects Hilbertian fields with ω-free fields.
Theorem. Let K be a PAC field. Then K is Hilbertian if and only if K is ω-free.
Peter Roquette proved the right-to-left direction of this theorem and conjectured the opposite direction. Michael Fried and Helmut Völklein applied algebraic topology and complex analysis to establish Roquette's conjecture in characteristic zero. Later Pop
proved the Theorem for arbitrary characteristic by developing "rigid patching".