Abstract

David Hilbert [Hil92] showed that for an irreducible polynomial F (X, T ) ∈ ℚ(T )[X] there are infinitely many rational numbers t for which F (X, t) is irreducible in ℚ[X]. In 1936 van der Waerden [vdW34] gave a quantitative form of this assertion. Consider the set of degree n monic polynomials with integer coefficients restricted to a box |a_i| ≤ B. Van der Waerden showed that a polynomial drawn at random from this set has Galois group Sn with probability going to 1 as B tends to infinity.

In the first part of the thesis, we introduce the Large Sieve Method and apply it to solve Probabilistic Galois The- ory problems over rational numbers. We estimate, E_n(B), the number of polynomials of degree n and height at most B whose Galois group is a proper subgroup of the symmetric group Sn. Van der Waerden conjectured that E_n(B) ≪ B^(n-1). P.X. Gallagher [Gal73] utilized an extension of the Large Sieve Method to obtain an estimate of E_n(B)= O(B^(n-1/2) log^(1-