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Gauss’s ternary form reduction and its application to a prime decomposition symbol

Published online by Cambridge University Press:  22 January 2016

Yoshiomi Furuta
Yoshiomi Furuta*
Affiliation:
Department of Mathematics, Kanazawa University, Kanazawa 920, Japan
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We defined a prime decomposition symbol [d1 d2, p] in a previous paper [3], and characterized in [4] the set of rational primes p which are decomposed completely in a non-abelian central extension which is of degree 8 in substance. The explicit value of the symbol was determined by using a solution of certain ternary quadratic diophantine equation. The solution corresponds to a square root of an ideal class of the principal genus of a quadratic field. This is translated to a problem in classes of integral quadratic forms, namely to find a form whose duplication is a given one contained in a principal genus. An explicit method to find the form is given by Gauss in [5, Art. 286], which is due to his ternary form reduction.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 98 , June 1985 , pp. 77 - 86
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1985

References

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