The large sieve is used to estimate the density of quadratic polynomials Q∈ℤx, such that there exists an odd degree polynomial defined over ℤ which has resultant ±1 with Q. Given a monic polynomial R∈ℤx of odd degree, this is used to show that for almost all quadratic polynomials Q∈ℤx, there exists a prime p such that Q and R share a common root in 𝔽 ¯ p . 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.
Browning et al. (Tue,) studied this question.