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A Kummer theoretic construction of an \(S_3\)-polynomial with given quadratic subfield. (English) Zbl 1262.11094

Following the book of C. U. Jensen, A. Ledet and N. Yui [Generic polynomials. Constructive aspects of the inverse Galois problem. Cambridge: Cambridge University Press (2002; Zbl 1042.12001)], the polynomial \(g(X)= X^3+ sX+ s\in\mathbb Q(s)[X]\) is generic for the symmetric group \(S_3\). In the opinion of the author, the arithmetic of this polynomial is not easy to deduce from the expression itself. By using a computer algebra system, this short paper provides an \(S_3\)-polynomial whose arithmetic is closely related to the chosen parameters. It arises from a Kummer theory of certain algebraic tori.

MSC:

11R32 Galois theory

Citations:

Zbl 1042.12001
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