# Field arithmetic

Field arithmetic

In 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 groups and 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

:$G=Gal\left(mathbf\left\{C\right\}/mathbf\left\{R\right\}\right)=mathbf\left\{Z\right\}/2 mathbf\left\{Z\right\}.$

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 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

:$hat mathbf\left\{Z\right\}=lim_\left\{longleftarrow\right\}mathbf\left\{Z\right\}/n mathbf\left\{Z\right\}.,$

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(("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 to Adrien Douady for 0 characteristic and has its origins in Riemann's existence theorem. For a field of arbitrary characteristic it is due to David Harbater and Florian Pop, and was also proved later by Dan Haran and 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 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 and Koji 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. 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 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 and Helmut 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 called rigid 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.

Wikimedia Foundation. 2010.

Нужна курсовая?

### Look at other dictionaries:

• Finite field arithmetic — Arithmetic in a finite field is different from standard integer arithmetic. There are a limited number of elements in the finite field; all operations performed in the finite field result in an element within that field.While each finite field is …   Wikipedia

• Arithmetic — tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word ἀριθμός, arithmos “number”) is the oldest and most elementary branch of mathematics, used b …   Wikipedia

• Field of definition — In mathematics, the field of definition of an algebraic variety V is essentially the smallest field to which the coefficients of the polynomials defining V can belong. Given polynomials, with coefficients in a field K , it may not be obvious… …   Wikipedia

• Arithmetic dynamics — is a new field that is an amalgamation of two areas of mathematics, dynamical systems and number theory. The subject can be viewed as the transfer of previous results in the theory of Diophantine equations to the setting of discrete dynamical… …   Wikipedia

• field — [fēld] n. [ME feld < OE, akin to Ger feld, Du veld < IE * pelt < base * pele , * pla , flat and broad > L planus, plane, Gr palamē, flat hand] 1. a wide stretch of open land; plain 2. a piece of cleared land, set off or enclosed, for… …   English World dictionary

• Field (mathematics) — This article is about fields in algebra. For fields in geometry, see Vector field. For other uses, see Field (disambiguation). In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it …   Wikipedia

• Arithmetic function — In number theory, an arithmetic (or arithmetical) function is a real or complex valued function ƒ(n) defined on the set of natural numbers (i.e. positive integers) that expresses some arithmetical property of n.  An example of an arithmetic… …   Wikipedia

• Arithmetic mean — In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space. The term arithmetic mean is preferred in mathematics and… …   Wikipedia

• Arithmetic of abelian varieties — In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or family of those. It goes back to the studies of Fermat on what are now recognised as elliptic curves; and has become a very… …   Wikipedia

• Arithmetic group — In mathematics, an arithmetic group (arithmetic subgroup) in a linear algebraic group G defined over a number field K is a subgroup Γ of G ( K ) that is commensurable with G ( O ), where O is the ring of integers of K . Here two subgroups A and B …   Wikipedia