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 November 16th, 2006, 08:14 AM #1 Newbie   Joined: Nov 2006 Posts: 28 Thanks: 0 Pseudo-finite fields How to show that pseudo-finite fields are the infinite models of the theory of finite fields? March 5th, 2009, 03:20 PM #2 Newbie   Joined: Mar 2009 Posts: 1 Thanks: 0 Re: Pseudo-finite fields Pseudo-finite fields are by definition infinite models of the theory of finite fields. One can take a field theoretic definition of pseudo-finite fields as well. It goes as follows: A field is pseudo-finite if it is perfect, pseudo algebraically closed and it has a unique extension of degree n for every n>0. Of course the first and the second definitions are equivalent. To show that a field which is pseudo-finite (i.e perfect, pseudo algebraically closed and it has a unique extension of degree n for every n>0) is an infinite model of the theory of finite fields one has to show that all these three properties are first order expressible. This is not trivial but can be done. Then one shows that these first order statements corresponding to the three properties are satisfied by all large enough finite fields, i.e. they hold in the theory of finite fields. Finally we have to show that a field satisfying the three properties has to be infinite. This comes from pseudo algebraically closedness. Finite fields are not pseudo algebraically closed, hence the only fields which satisfy the three properties are the infinite models of the theory of finite fields. (Which do exist because of the compactness, the theory of finite fields have arbitrarily large models, therefore they have infinite models as well.) Tags fields, pseudofinite Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post gelatine1 Abstract Algebra 2 February 1st, 2013 03:03 PM gelatine1 Number Theory 0 November 7th, 2012 08:44 AM lreps Abstract Algebra 1 April 12th, 2011 02:07 PM lreps Abstract Algebra 0 April 10th, 2011 01:38 PM lreps Number Theory 0 December 31st, 1969 04:00 PM

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