Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: The Degree of the Splitting Field of a Random Polynomial over a Finite Field. | The Degree of the Splitting Field of a Random Polynomial over a Finite Field John D. Dixon and Daniel Panario School of Mathematics and Statistics Carleton University Ottawa Canada jdixon daniel @ Submitted Aug 30 2004 Accepted Sep 22 2004 Published Sep 30 2004 Mathematics Subject Classifications 11T06 20B99 Abstract The asymptotics of the order of a random permutation have been widely studied. P. Erdos and P. Turan proved that asymptotically the distribution of the logarithm of the order of an element in the symmetric group Sn is normal with mean 2 log n 2 and variance 3 log n 3. More recently R. Stong has shown that the mean of the order is asymptotically exp Cựn log n O ựn log log n log n where C . We prove similar results for the asymptotics of the degree of the splitting field of a random polynomial of degree n over a finite field. 1 Introduction We consider the following problem. Let F denote a finite field of size q and consider the set Pn q of monic polynomials of degree n over F. What can we say about the degree over F of the splitting field of a random polynomial from Pn q Because we are dealing with finite fields and there is only one field of each size it is well known that the degree of the splitting field of f X G Pn q is the least common multiple of the degrees of the irreducible factors of f X over F. Thus the problem can be rephrased as follows. Let A be a partition of n denoted A b n and write A in the form where A has ks parts of size s. We shall say that a polynomial is of shape A if it has ks irreducible factors of degree s for each s. Let w A q be the proportion of polynomials in Pn q which have shape A. If we define m A to be the least common multiple of the sizes of the parts of A then the degree of the splitting field over F of a polynomial of shape A is m A . The average degree of a splitting field is given by E q 22w A q m A . x -n THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R70 1 An analogous problem
