Re: Computational aspects of Galois theory?

*To*: mathgroup at smc.vnet.net*Subject*: [mg43728] Re: [mg43694] Computational aspects of Galois theory?*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Thu, 2 Oct 2003 02:51:27 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Your statement of the problems seems strange to me and I am not referring to the fact that it is written in TeX, while this is a Mathematica mailing list ;-) . If we take your description of the problem at face value the field in question seems to be just the rationals, since the set of all rational combinations of a set of rationals is trivially the rationals. I am sure you must mean something else, but at this moment I can't guess what. Andrzej Kozlowski Yokohama, Japan http://www.mimuw.edu.pl/~akoz/ http://platon.c.u-tokyo.ac.jp/andrzej/ On Wednesday, October 1, 2003, at 05:42 AM, Victor Alexandrov wrote: > Let $p_1$,..., $p_n$ be $n$ different prime numbers. > For every subset $A$ of the set $\{ 1,...,n\}$ consider > the product $q_A=\prod_{i\in A} p_i$. If $A$ is empty > $q_A$ equals 1 by definition. The set of all linear > combinations of $q_A$ with rational coefficients is > denoted by $F(1,p_1,...,p_n)$. It is well-known that > $F(1,p_1,...,p_n)$ is a field. > Does there exist a software which, for a given number > $x$ of $F(1,p_1,...,p_n)$, says whether the square root > of $x$ belongs to $F(1,p_1,...,p_n)$ and, when possible, > represents this square root as a linear combination of > $q_A$ with rational coefficients? > > >