Solubility of a resultant equation and applications

The large sieve is used to estimate the density of quadratic polynomials <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>ℤ</mml:mi> <mml:mo>[</mml:mo> <mml:mi>x</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> , such that there exists an odd degree polynomial defined over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℤ</mml:mi> </mml:math> which has resultant <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>±</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> . Given a monic polynomial <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>ℤ</mml:mi> <mml:mo>[</mml:mo> <mml:mi>x</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> of odd degree, this is used to show that for almost all quadratic polynomials <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>ℤ</mml:mi> <mml:mo>[</mml:mo> <mml:mi>x</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> , there exists a prime <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> such that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> share a common root in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>𝔽</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>p</mml:mi> </mml:msub> </mml:math> . Using recent work of Landesman, an application to the average size of the odd part of the class group of quadratic number fields is also given.