Abstract
Hilbert’s Irreducibility Theorem is applied to find the upper bounds of the time complexities of various decision problems in arithmetical sentences and the following results are proved: 1. The decision problem of ∀ ∃ sentences over an algebraic number field is in P. 2. The decision problem of ∀ ∃ sentences over the collection of all fields with characteristic 0 is in P. 3. The decision problem of ∀ ∃ sentences over a function field with characteristic p is polynomial time reducible to the factorization of polynomials over Z p . 4. The decision problem of ∀ ∃ sentences over the collection of all fields with characteristic p is polynomial time reducible to the factorization of polynomials over Z p . 5. The decision problem of ∀ ∃ sentences over the collection of all fields is polynomial time reducible to the factorization of integers over Z and the factorization of polynomials over finite fields.
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