Number Theory

Read e-book online An Introduction to Number Theory 2 DVD Set with Guidebook PDF

By Edward B. Burger

2 DVD set with 24 lectures half-hour each one for a complete of 720 minutes...

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9 the relation ~ in K is elementarily definable 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.

Let :F be a non-principal ultrafilter on N and let K* be the ultraproduct IlnENKn/ :F of the above sequence of fields with respect to a non-principal ultrafilter :Fon N. Since Kn ~ R, each Kn is formally real and therefore K* is formally real. The property that for every a either a or -a is a square holds in each Kn and therefore also in K*. Finally, every polynomial of odd degrees in Kn[X] has a root in Kn for all n > s. Consequently, each polynomial in K*[X] of degrees has a root in K*. By ArtinSchreier's theorem K* is real closed.

To verify that the real closed fields do not form a finitely axiomatizable class we need: 22 CHAPTER 2. 25 For any natural numbern there is a subfield Kn of the real number field R such that ( 1) Kn is not real closed. (2) For each positive number a in Kn the square root Va belongs to Kn· (3) Every polynomial in Kn[X] of odd degree n has a root in Kn· Proof. Let F0 , F11 F2, ... be the sequence of number fields defined inductively as follows: Let F0 be the rational number field Q. Let Ft+l be the field obtained from Ft by adjoining all square roots of positive numbers in Fi and all real algebraic numbers whose degree relative to Ft is an odd number ::; n.

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