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Computational aspects of Galois theory?

  • To: mathgroup at
  • Subject: [mg43694] Computational aspects of Galois theory?
  • From: alex at (Victor Alexandrov)
  • Date: Tue, 30 Sep 2003 16:42:46 -0400 (EDT)
  • Sender: owner-wri-mathgroup at

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?

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