From: Edwin Clark Subject: Re: Galois group of random polynomial Date: Sat, 10 Apr 1999 18:52:32 -0400 Newsgroups: sci.math.research I received quite a few replies to my query about whether or not a "random" polynomial over Q (1) is irreducible in Q[x], (2) has Galois group S_n over Q. Although there has been some posted discussion of what is meant by random, I haven't seen the following references. (I don't know if they were posted and have not appeared yet, or if they were just sent to me.) Anyhow here they are if anyone is interested. [Note to moderator: It would be nice if all postings were placed on some website where one could go to find a complete and up-to-date archive.] -------------------------------------------------- Douglas Zare pointed out that (2) => (1). So if one is only interested in the end result it suffices to establish (2). He also sketched a proof of (2). -------------------------------------------------- The oldest reference mentioned was supplied by Wladyslaw Narkiewicz. He writes that both were proved by B.L.van der Waerden in Mathematische Annalen, 109, 1931, page 13. -------------------------------------------------- Dani Berend sends the following also by van der Waerden: \item{[vW]} B. L. van der Waerden, Die Seltenheit der Gleichungen mit Affekt, {\it Math. Ann.} {\bf 109} (1933), 13--16. -------------------------------------------------- Igor Shparlinski furnishes the following more recent references on the subject: \ref \by R. Chela \paper Reducible polynomials \jour J. London Math. Soc. \yr 1963 \vol 38 \pages 183--188 \endref \ref \by S. D. Cohen \paper The distribution of Galois groups of integral polynomials \jour Illinois J. of Math. \yr 1979 \vol 23 \pages 135--152 \endref \ref \by S. D. Cohen \paper The distribution of Galois groups and Hilbert's irreducibility theorem \jour Proc. London Math. Soc. \yr 1981 \vol 43 \pages 227--250 \endref -------------------------------------------------- Pieter Moree sent the following: See the beginning of J.-P. Serre, Topics in Galois theory. Another paper that might interest you is by Davis et al., Probabilistic Galois theory of reciprocal polynomials, Expositiones Mathematicae 16 (1998), 263-270. On a slightly different note there is the classic result of Schur that if you cut off the Taylor series for e^x at x^n, then the resulting polynomial over Q has Galois group S_n. For more recent work in this direction there are some papers of M. Filaseta; 97g:11025 and 97b:11034. -------------------------------------------------- Thanks to all who responded! --Edwin Clark ------------------------------------------------------ W. Edwin Clark Department of Mathematics, University of South Florida http://www.math.usf.edu/~eclark/ ------------------------------------------------------ ============================================================================== From: igor@mpce.mq.edu.au (Igor Shparlinski) Subject: Re: Galois group of random polynomial Date: 9 Apr 99 12:22:47 GMT Newsgroups: sci.math.numberthy At 3:44 PM +1100 9/4/99, Edwin Clark wrote: >I somehow have the idea that a random polynomial >of degree n with coefficients in the rationals (Q) >with probability 1 as n -> infinity > >(1) is irreducible over Q, > >and > >(2) has galois group over Q isomorphic to >the symmetric group S_n. > >Can someone provide references or arguments to >support these assertions (assuming they are correct.) Hi, Yes (1) and (2) are correct and well known. In more precise forms (with estimates on the number of exceptions) they can be found: (1) in, say, \ref \by R. Chela \paper Reducible polynomials \jour J. London Math. Soc. \yr 1963 \vol 38 \pages 183--188 \endref (2) in, say, \ref \by P. X. Gallagher \paper The large sieve and probabilistic Galois theory \jour Proc. Symp. in Pure Math. \publ AMS \yr 1973 \vol 24 \pages 91--101 \endref Even more precise results have been obtained by Steve Cohen, e.g., \ref \by S. D. Cohen \paper The distribution of Galois groups of integral polynomials \jour Illinois J. of Math. \yr 1979 \vol 23 \pages 135--152 \endref \ref \by S. D. Cohen \paper The distribution of Galois groups and Hilbert's irreducibility theorem \jour Proc. London Math. Soc. \yr 1981 \vol 43 \pages 227--250 \endref Pobably there are some later publications as well. Best wishes, Igor +------------------------------------------------------------------+ | Dr Igor Shparlinski | | | | E-mail: igor@comp.mq.edu.au | | Department of Computing | Ph.: [61-(0)2] 9850 9585 | | Macquarie University | FAX: [61-(0)2] 9850 9551 | | NSW 2109, | Home page: | | Australia | http://www.comp.mq.edu.au/~igor | +------------------------------------------------------------------+