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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?
>
>
>