Jeroen Demeyer, Jan Van Geel: An Existential Divisibility Lemma for Global Fields

Submission: 2004, Jun 16

Let K be a global field of characteristic not 2 and L a quadratic extension of K. We prove that there exists an existential formula which holds true for elements x,y in K if and only if for all primes p of K inert in L for which x has odd valuation it follows that the valuation of x is greater than 2 times the valuation of y. Such existential formulas play a role in strategies to obtain undecidability results for the existential theory of K. The proof is based on the Hasse principle for quadratic forms, the (strong) approximation theorem and Hilbert's reciprocity law.

2000 Mathematics Subject Classification: 11E12 11R11, 11U05, 03B25

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