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Galois groups of prime degree polynomials with nonreal roots. (English) Zbl 1134.12002

Shaska, Tanush (ed.), Computational aspects of algebraic curves. Papers from the conference, University of Idaho, Moscow, ID, USA, May 26–28, 2005. Hackensack, NJ: World Scientific (ISBN 981-256-459-4/hbk). Lecture Notes Series on Computing 13, 243-255 (2005).
Computing Galois group of polynomials is a largely open difficult problem. Even with the development of new computer algebra systems, this remains a challenge and can be carried out only for small degree polynomials (currently \(\leq 15)\). In this note, the authors study the case of polynomials, of prime degree \(p\), with nonreal roots. The existence of nonreal roots makes the computation of its Galois group much easier since numerical methods can be used. Indeed, it follows from a theorem of Jordan (1871) that if the number of nonreal roots is “small” enough with respect to the degree \(p\), then the Galois group is \(A_p\) or \(S_p\). Precisely, let \(f(x)\in\mathbb{Q}[x]\) be an irreducible polynomial of prime degree \(p\geq 5\) and \(2s\) be the number of nonreal roots of \(f(x)\). It is proved that if \(s\) satisfies \(s(s\log(s)+2\log(s)+3)\leq p\), then \(\text{Gal} (f)=A_p\), \(S_p\). When this inequality does not hold, some exceptional groups can occur. Using GAP, the authors set the table in that case, of all possible Galois groups of polynomials, of prime degree \(p\leq 29\), with nonreal roots.
For the entire collection see [Zbl 1109.14003].

MSC:

12F10 Separable extensions, Galois theory
12-04 Software, source code, etc. for problems pertaining to field theory
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