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On small discriminants of number fields of degree 8 and 9. (English. French summary) Zbl 1460.11127

Two fundamental problems in algebraic number theory are the determination of the smallest absolute value of discriminant of a number field with given signature \((r_1,r_2)\), and the classification of all those fields with a given discriminant. A complete classification of number fields of degree at the most \(7\), and up to a certain discriminant are known using diverse methods.
The author – in his PhD thesis – gave a complete classification of number fields with degree \(8\), signature \((2,3)\) and discriminant bounded by \(5726301\), showing that there exist exactly \(56\) such fields. In the present paper, the author extends this result to the fields of degree \(8\) and signatures \((4,2)\) and \((6,1)\), and to number fields of degree \(9\) and signatures \((1,4)\) and \((3,3)\). The precise result is the following.
There exist \(41\) number fields with signature \((4,2)\) and absolute value of discriminant not exceeding \(20829049\). Moreover, the minimum value of the absolute value of discriminant for the signature \((4,2)\) is \(15243125.\)
There exist \(8\) number fields with signature \((6,1)\) and absolute value of discriminant not exceeding \(79259702\). Moreover, the minimum value of the absolute value of discriminant for this signature is \(65106259\).
There exist \(67\) number fields with signature \((1,4)\) and absolute value of discriminant not exceeding \(39657561\). Moreover, the minimum value of the absolute value of discriminant for this signature is \(29510281\).
There exist \(116\) number fields with signature \((3,3)\) and absolute value of discriminant not exceeding \(146723910\). Moreover, the minimum value of the absolute value of discriminant for this signature is \(109880167\).

MSC:

11R21 Other number fields
11R29 Class numbers, class groups, discriminants
11Y40 Algebraic number theory computations

Software:

PARI/GP; LMFDB
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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