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Index form equations in quintic fields. (English) Zbl 0930.11091

This paper gives an improved method to solve index form equations, with examples given to illustrate the case of quintic fields with large Galois group, such as \(S_5\). Such fields have up to now been out of reach of modern methods.
The method involves reduction to a set of two term unit equations with a restricted set of exponential variables. This is done using a careful analysis of which fields the units involved belong to. The unit equations are then solved to give the solutions to the index form equation.
The method to solve the unit equation starts in the standard way. First astronomically large upper bounds are produced using the theory of linear forms in logarithms. Then these upper bounds are reduced using the LLL algorithm. The final enumeration of the “small” solutions is performed using a modification of Wildanger’s method, which is itself based on the enumeration of lattice points in certain ellipsoids and hence uses the Fincke-Pohst algorithm.

MSC:

11Y50 Computer solution of Diophantine equations
11D57 Multiplicative and norm form equations
11R21 Other number fields
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