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
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