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Computing all power integral bases of cubic fields. (English) Zbl 0677.10013
Let K be a cubic number field with integral basis \(\{w_ 1=1,w_ 2,w_ 3\}\). Define the index form I(x,y) of K by \(D_{K/{\mathbb{Q}}}(w_ 2x+w_ 3y)=(I(x,y))^ 2D,\) where D is the discriminant of K and \(D_{K/{\mathbb{Q}}}(a)\) the discriminant of \(a\in K\). \(\{1,a,a^ 2\}\) is a power integral basis of K if and only if \(D_{K/{\mathbb{Q}}}(a)=D\). Hence we can find all power integral bases of K by solving the index form equation (1) \(I(x,y)=\pm 1\) in \(x,y\in {\mathbb{Z}}.\)
The authors solved all equations (1), up to \(GL_ 2({\mathbb{Z}})\)- equivalence, for -300\(\leq D\leq 3137\). First they computed a large upper bound for the solutions of (1) by means of Baker’s theory on linear forms in logarithms; then they reduced this bound to a much smaller one, by means of a lemma of Baker-Davenport, and then they computed the small solutions, using a method of Pethö. Only for \(D=49\) there were nine solutions; for \(D=81\), 229, 257, 361 there were six solutions and in all other cases at most five.
Reviewer: J.H.Evertse

11D25 Cubic and quartic Diophantine equations
11-04 Software, source code, etc. for problems pertaining to number theory
11D57 Multiplicative and norm form equations
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