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By Pierluigi Crescenzi, Viggo Kann.

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Extra info for A Compendium of NP Optimization Problems

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This resolvent is the well-known degree 6 resolvent. We see that: 1. if g factors as a linear times a quintic, then either G D C5 , D5 , or F20 . 2. if g remains irreducible, then G D A5 or S5 . , the resolvent corresponding to A5 ). The discriminant also Absolute Resolvents and Masses of Irreducible Quintic Polynomials 37 determines when G D F20 . To distinguish between C5 and D5 , we can use the resolvent corresponding to the group C3 C2 , and this is the smallest absolute resolvent that accomplishes that purpose.

If g remains irreducible, then G D A5 or S5 . , the resolvent corresponding to A5 ). The discriminant also Absolute Resolvents and Masses of Irreducible Quintic Polynomials 37 determines when G D F20 . To distinguish between C5 and D5 , we can use the resolvent corresponding to the group C3 C2 , and this is the smallest absolute resolvent that accomplishes that purpose. We point out that Cohen makes use of a different method to distinguish between C5 and D5 , which is based on Stauduhar’s relative resolvent method [15].

Most modern treatments employ a degree 6 resolvent polynomial to determine the Galois group [9, 10]. When this resolvent is factored over the base field, two scenarios can occur. In one case, the resolvent remains irreducible, which indicates the Galois group is either A5 or S5 , C. R. edu © Springer International Publishing Switzerland 2015 J. Rychtáˇr et al. 1007/978-3-319-11125-4__4 31 32 C. R. Shill and therefore the original quintic polynomial is not solvable by radicals. In the other case, the degree 6 resolvent factors as a linear times an irreducible quintic, which indicates the Galois group is either cyclic, dihedral, or the previously mentioned metacyclic group of order 20.

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A Compendium of NP Optimization Problems by Pierluigi Crescenzi, Viggo Kann.


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