Date: Fri, 7 Jun 96 15:49:37 CDT
From: rusin (Dave Rusin)
To: daw@cs.anu.edu.au
Subject: Re: Irreducibility criteria
Newsgroups: sci.math.research
In article <31B7BFB1.54F9@cs.anu.edu.au> you write:
>Magma tells me that
>
>x^k - x^{k-1} - ... - x^2 - x - 1
>
>is irreducible over the rationals where 1 <= k <= 865.
>
>Does someone know of a criterion other than Eisenstein's for
>showing that it is irreducible in general?
You want to know if x^(k+1) - 2 x^k + 1 has any factors besides
x-1. (It may be nicer to substitute 1/x for x so that you have
to look at x^(k+1) - 2 x + 1.) Well, I don't know if there ever are
other factors, but there are papers on just this sort of thing
(e.g. replace the "2" by "1" and it's known). Here's a URL:
http://www.math.niu.edu/~rusin/papers/known-math/95/trinom
[URL updated 1999/01 -- djr]
Let me know what you eventually decide. I could have sworn I'd seen this
problem before but I've lost references to it.
dave
==============================================================================
From: David A Wolfram
Subject: Re: Irreducibility criteria
To: rusin@math.niu.edu (Dave Rusin)
Date: Thu, 13 Jun 1996 11:04:08 +1000 (EST)
[previous letter quoted -- djr]
Hi,
I have taken a browse and Osada's paper is relevant for another
question. Thanks for mentioning this URL. David Boyd told me that the
polynomial crops up in the theory of Pisot numbers and its
irreducibility was proved in that context. Exactly k-1 of the roots of
the k^th order polynomial have moduli strictly less than one. If it
were reducible, the product of some of these roots would have to be an
integer which is not possible.
Regards,
David W.