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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]].
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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==
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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  
 
:<math>G=Gal(\mathbf{C}/\mathbf{R})=\mathbf{Z}/2 \mathbf{Z}.</math>
 
Here '''C''' is the field of complex numbers and '''Z''' is the ring of integer numbers.  
A [[Artin–Schreier theorem|theorem of Artin and 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'''/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
 
:<math>\hat{\mathbf{Z}}=\lim_{\longleftarrow}\mathbf{Z}/n \mathbf{Z}.\,</math>  
 
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 ''N<sub>s</sub>'' be the maximal Galois extension of '' '''Q''' '' that ''s'' fixes. Then with probability 1 the absolute Galois group Gal(''N''<sub>''s''</sub>) 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'':&nbsp;Gal(''K'')&nbsp;→&nbsp;Gal(''L'') be an isomorphism. Then there exists a unique isomorphism of the algebraic closures, ''b'': ''K''<sub>alg</sub>&nbsp;→&nbsp;''L''<sub>alg</sub>, that induces  ''a''.
 
This generalizes an earlier work of [[Jürgen Neukirch]] and [[Koji Uchida]] on number fields.
 
==Pseudo algebraically closed fields==
{{Main|Pseudo algebraically closed field}}
 
A [[pseudo algebraically closed field]] (in short PAC) ''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 field]]s with ω-free fields (''K'' is ω-free if any [[embedding problem]] for ''K'' is properly 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. [[Michael Fried (mathematician)|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]]".
 
==References==
{{reflist}}
* {{cite book | last1=Fried | first1=Michael D. | last2=Jarden | first2=Moshe | title=Field arithmetic | edition=2nd revised and enlarged | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge | volume=11 | publisher=[[Springer-Verlag]] | year=2004 | isbn=3-540-22811-X | zbl=1055.12003 }}
*{{Neukirch et al. CNF}}
 
[[Category:Algebra]]
[[Category:Galois theory]]

Latest revision as of 23:45, 16 September 2014

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