Defining the integers in large rings of number fields using one universal quantifier
نویسنده
چکیده
Julia Robinson has given a first-order definition of the rational integers Z in the rational numbers Q by a formula (∀∃∀∃)(F = 0) where the ∀quantifiers run over a total of 8 variables, and where F is a polynomial. We show that for a large class of number fields, including Q, for every ε > 0, there exists a set of primes S of natural density exceeding 1− ε, such that Z can be defined as a subset of the “large” subring {x ∈ K : ordp x > 0, ∀ p 6∈ S } of K by a formula of the form (∃∀∃)(F = 0) where there is only one ∀-quantifier, and where F is a polynomial.
منابع مشابه
Defining the integers in large rings of a number field using one universal quantifier
Julia Robinson has given a first-order definition of the rational integers Z in the rational numbers Q by a formula (∀∃∀∃)(F = 0) where the ∀quantifiers run over a total of 8 variables, and where F is a polynomial. We show that for a large class of number fields, not including Q, for every ε > 0, there exists a set of primes S of natural density exceeding 1− ε, such that Z can be defined as a s...
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