Journal of the Ramanujan Mathematical Society

Yu-Ru Liu ¹, M. Ram Murty ²

Affiliations
1 Department of Mathematics, Harvard University Cambridge, Mass., 02138, US
2 Department of Mathematics and Statistics, Queen's University, Kingston Ontario, K7L 3N6, CA

Abstract

We introduce the Turan sieve method and apply it to the probabilistic Galois theory problems in both the rational number field and the function field cases. We estimate the number of polynomials of degree n and height ≤ N whose Galois group is a proper subgroup of Sn- For the rational number field case, we get an estimate of O(N^n-1/3(logN)<sup.2) and in the case of the function field over F_q, we get O(N^n-1log_qN).

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