Number Theory

Model Theoretic Algebra With Particular Emphasis on Fields, by Christian. U Jensen

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By Christian. U Jensen

This quantity highlights the hyperlinks among version concept and algebra. The paintings encompasses a definitive account of algebraically compact modules, a subject of imperative significance for either module and version thought. utilizing concrete examples, specific emphasis is given to version theoretic ideas, equivalent to axiomizability. natural mathematicians, in particular algebraists, ring theorists, logicians, version theorists and illustration theorists, should still locate this an soaking up and stimulating booklet.

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14. p E P which is an isomorphism from a subfield I<' of I< to a subfield L' of L, we define an isomorphism t:p: I<'[{X;}] ~ L'[{X;}] (resp. I<'( {X;}) ~ L'( {X;} )) in the obvious coefficentwise way. 14 D for the rings I<[{X;}] and L[{X;}], resp. the fields I<({X;}) and L({X;}). 20 Let K and L be algebraically closed fields of characteristic zero and {X;};e1 and {l'ilieJ two non-empty {finite or infinite} families of indeterminates. degq(L) are either both infinite or both finite and equal. JI.

P'. Then R and S are elementarily equivalent. 40 CHAPTER Proof. xi, . . , Xn 3: ELEMENTARY DEFINABILITY We use the fact that every first order sentence a with variables can be expressed in the form where each Q; is a quantifier, either 3 or V, and t/; is a formula in xi, ... , Xn, without quantifiers. By allowing vacuous occurrences of quantifiers we may assume that a has the form where t/; is a formula in Xi, yi, . , Xn, Yn without quantifiers. Suppose a holds in S. We must show that a also holds in R.

24 that the statement 'V has a prime ideal of height t', for a prescribed integer t, will hold true if and only if a certain first order sentence holds in R. 26 A modification of the above proof shows that if /{ and L are real closed fields and then the prime ideals of VK form a dense totally ordered set if and only if the prime ideals of VL form a dense totally ordered set. e. K-dim(VFn)) is n - 1, hence for any set of indeterminates {X;} the polynomial rings are pairwise elementarily inequivalent.

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