NetNews Usenet Archive 1992 #26/NN_1992_26.iso/spool/sci/math/14644

Internet Message Format | 1992-11-09 | 1.8 KB

Path: sparky!uunet!mozz.unh.edu!kepler.unh.edu!dvf
From: [e-mail] (David V Feldman)
Newsgroups: sci.math
Subject: Re: Univariate polynomial equations and the FAQ
Date: 9 Nov 1992 05:12:34 GMT
Organization: University of New Hampshire  -  Durham, NH
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In article <[e-mail]> [e-mail] (Eugen W. Schmidt) writes:
>In article <[e-mail]> [e-mail] (David V Feldman) writes:
>
>>Fix an integer m.  Let K be the extension of Q obtained by adjoining
>>all roots of all polynomials of the form
>>     n     m
>>      x + a x + ... a
>>           m         0
>
>>where the coefficients are rational.
>
>This sounds like the algebraic closure of Q, Q-bar.

While it is true that I did not ask the question that I intended to,
I don't see any reason that  K = Q-bar.  Remember that  m  is fixed.
So I was adjoining roots of polynomials that, for large  n , would
have very few non-zero coefficients.  Actually the field  K  that
I intend is even larger.  Specifically, what I should have said is:

Fix an integer  m .  Let K be the smallest extension of Q closed under
the adjunction of all roots of all polynomials of the form

       n     m
      x + a x + ... a
           m         0

with the a_i in K .  Then let s(m) be the smallest degree of a polynomial
with coefficients in  Q  and no root in K.  Then if  m=0, s(0)=5 by
Galois theory.  Again, the question is, what is known about s(m), m>0?

>   Eugen W. Schmidt/Der Brahms Gang/IWR/Ruprecht-Karls University
>               [e-mail]

David Feldman