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Sieve Methods and its Application in Probabilistic Galois Theory


Sieve Methods and its Application in Probabilistic Galois Theory

Bahar, Salik (2017) Sieve Methods and its Application in Probabilistic Galois Theory. Masters thesis, Concordia University.

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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-

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Thesis (Masters)
Authors:Bahar, Salik
Institution:Concordia University
Degree Name:M.A.
Date:27 September 2017
Thesis Supervisor(s):David, Chantal
ID Code:983088
Deposited By: SALIK BAHAR
Deposited On:17 Nov 2017 16:35
Last Modified:18 Jan 2018 17:56
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