- Field arithmetic
In

mathematics ,**field arithmetic**is a subject that studies the interrelations between arithmetic properties of a ql|field_(mathematics)|field and itsabsolute Galois group .It is an interdisciplinary subject as it uses tools fromalgebraic number theory ,arithmetic geometry ,algebraic geometry ,model theory , the theory offinite groups and ofprofinite groups .**Fields with finite absolute Galois groups**Let "K" be a field and let "G" = Gal("K") be its absolute Galois group. If "K" is

algebraically closed , then "G" = 1". If "K" =**R**is the real numbers, then:$G=Gal(mathbf\{C\}/mathbf\{R\})=mathbf\{Z\}/2\; mathbf\{Z\}.$

Here

**C**is the field of complex numbers and**Z**is the ring of integer numbers. A theorem of Artin-Schreier asserts that (essentially) 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" isreal closed and "G" =**Z**/2**Z**.**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

:$hat\; mathbf\{Z\}=lim\_\{longleftarrow\}mathbf\{Z\}/n\; mathbf\{Z\}.,$

This group is isomorphic to the absolute Galois group of an arbitrary

finite field . Also the absolute Galois group of the field offormal Laurent series **C**(("t")) 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 (that is

free profinite group ).* Let "C" be an

algebraically closed field and "x" a variable. Then Gal("C"("x")) is free of rank equal to the cardinality of "C". (This result is due toAdrien Douady for 0 characteristic and has its origins inRiemann's existence theorem . For a field of arbitrary characteristic it is due toDavid Harbater andFlorian Pop , and was also proved later byDan Haran andMoshe Jarden .)* The absolute Galois group Gal(

**Q**) (where**Q**are the rational numbers) is compact, and hence equipped with a normalizedHaar measure . For a Galois automorphism "s" (that is an element in Gal(**Q**)) let "N_{s}" be the maximal Galois extension of "**Q**" that "s" fixes. Then with probability 1 the absolute Galois group Gal("N"_{"s"}) is free of countable rank. (This result is due toMoshe Jarden .)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("K") → Gal("L") be an isomorphism. Then there exists a unique isomorphism of the algebraic closures, "b": "K"_{alg}→ "L"_{alg}, that induces "a".This generalizes an earlier work of

Jurgen Neukirch andKoji Uchida on number fields.**Pseudo algebraically closed fields**A

pseudo algebraically closed field (in short PAC) "K" is a field satisfying the following geometric feature. Eachabsolutely 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 field with ω-free fields ("K" is ω-free if any

embedding problem for "K" is solvable).**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.Michel Fried andHelmut Völklein applied algebraic topology and complex analysis to establish Roquette's conjecture in characteristic zero. Later Pop finished the job, andproved the Theorem for arbitrary characteristic, bydeveloping the so calledrigid patching .**References***M. D. Fried and M. Jarden, "Field Arithmetic", Springer-Verlag, Berlin, 2005.

*J. Neukirch, A. Schmidt, and K. Wingberg, "Cohomology of Number Fields", Springer-Verlag, Berlin Heidelberg, 2000.

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