- Field arithmetic
mathematics, field arithmetic is a subject that studies the interrelations between arithmetic properties of a ql|field_(mathematics)|field 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 groupsand of profinite 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
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" is
real closedand "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 seriesC(("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 closedfield and "x" a variable. Then Gal("C"("x")) is free of rank equal to the cardinality of "C". (This result is due to Adrien Douadyfor 0 characteristic and has its origins in Riemann's existence theorem. For a field of arbitrary characteristic it is due to David Harbaterand Florian Pop, and was also proved later by Dan Haranand Moshe Jarden.)
* The absolute Galois group Gal(Q) (where Q are the rational numbers) is compact, and hence equipped with a normalized
Haar measure. For a Galois automorphism "s" (that is an element in Gal(Q)) let "Ns" 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 to Moshe Jarden.)
In contrast to the above examples, if the fields in question are finitely generated over "Q",
Florian Popproves 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 Neukirchand Koji Uchidaon number fields.
Pseudo algebraically closed fields
pseudo algebraically closed field(in short PAC) "K" is a field satisfying the following geometric feature. Each absolutely irreduciblealgebraic 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 problemfor "K" is solvable).
Theorem. Let "K" be a PAC field. Then "K" is Hilbertian if and only if "K" is ω-free.
Peter Roquetteproved the right-to-left direction of this theorem and conjectured the opposite direction. Michel Friedand Helmut Völkleinapplied 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 called rigid patching.
*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.
Wikimedia Foundation. 2010.