It is shown that the compositum Q (2) of all degree 2 extensions of Q has undecidable theory.MSC: 11U05, 03B25, 11R11.
IntroductionIn this note we are interested in the following question. Problem 1. For which infinite algebraic extensions K of Q is the theory Th(K) undecidable? This question was first raised by A. Tarski and J. Robinson. In the 1930's A. Tarski showed that Q alg and R ∩ Q alg have decidable theories, and in 1959 J. Robinson showed that all number fields (that is, finite extensions of Q) have undecidable theory. Since there are uncountably many, non-elementarily equivalent, infinite algebraic extensions of Q and only countably many decision algorithms, it follows that most of them are undecidable. Such examples were pointed out by J. Robinson [4]: for any non-recursive set S of prime numbers the field Q S = Q({ √ p : p ∈ S}) has undecidable theory. Later the third named author [7] showed that the field Q S has undecidable theory for any infinite set of primes S.An interesting class of fields in which to study the above question, and to test current methods is the class of fields K (d) , which are the compositum of all extensions fields F/K of degree at most d over K, where K is a number field.These fields are Galois over K of infinite degree over K, and every element ofwhere S is the set of prime numbers that divide d!. E. Bombieri and U. Zannier [1] conjecture that these fields have the Northcott property making them, in this respect, similar to number fields. They proved that K (2) has the Northcott property.
In this note we study generic existence of maximal almost disjoint (MAD) families. Among other results we prove that Cohen-indestructible families exist generically if and only if b = c. We obtain analogous results for other combinatorial properties of MAD families, including Sacks-indestructibility and being +-Ramsey.
It is shown that the positive existential theory of the structure (ℤ[S−1]; =, 0, 1, + , |), where S is a nonempty finite set of prime numbers, is undecidable. This result should be put in contrast with the fact that the positive existential theory of (ℤ; =, 0, 1, + |) is decidable.
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