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This answers the original question. The modfied version is quite different.

No. The set of subsets of $P$ satisfying the OP conditions is of cardinality continuum while the set of polynomials with integer coefficients is countable. 

This answer gives implicite examples. But see this question, the answer of SashaP there and a comment by GHfromMO for concrete examples.

No. The set of subsets of $P$ satisfying the OP conditions is of cardinality continuum while the set of polynomials with integer coefficients is countable. This answer gives implicite examples. But see this question, the answer of SashaP there and a comment by GHfromMO for concrete examples.

This answers the original question. The modfied version is quite different.

No. The set of subsets of $P$ satisfying the OP conditions is of cardinality continuum while the set of polynomials with integer coefficients is countable. 

This answer gives implicite examples. But see this question, the answer of SashaP there and a comment by GHfromMO for concrete examples.

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No. The set of subsets of $P$ satisfying the OP conditions is of cardinality continuum while the set of polynomials with integer coefficients is countable. This answer gives implicite examples. But see this questionthis question, the answer of SashaP there and a comment by GHfromMO for concrete examples: Primes mod 4 and integer polynomials .

No. The set of subsets of $P$ satisfying the OP conditions is of cardinality continuum while the set of polynomials with integer coefficients is countable. This answer gives implicite examples. But see this question, the answer of SashaP there and a comment by GHfromMO for concrete examples: Primes mod 4 and integer polynomials .

No. The set of subsets of $P$ satisfying the OP conditions is of cardinality continuum while the set of polynomials with integer coefficients is countable. This answer gives implicite examples. But see this question, the answer of SashaP there and a comment by GHfromMO for concrete examples.

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No. The set of subsets of $P$ satisfying the OP conditions is of cardinality continuum while the set of polynomials with integer coefficients is countable. This answer gives implicite examples. But see this question, the answer of SashaP there and a comment by GHfromMO for concrete examples: Primes mod 4 and integer polynomials .

No. The set of subsets of $P$ satisfying the OP conditions is of cardinality continuum while the set of polynomials with integer coefficients is countable.

No. The set of subsets of $P$ satisfying the OP conditions is of cardinality continuum while the set of polynomials with integer coefficients is countable. This answer gives implicite examples. But see this question, the answer of SashaP there and a comment by GHfromMO for concrete examples: Primes mod 4 and integer polynomials .

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