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

  • Yoshiomi Furuta (a1)

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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.

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References

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[ 1 ] Cassels, J. W. S., Rational quadratic forms, Academic Press (1978).
[ 2 ] Dulin, B. J. and Butts, H. S., Composition of binary quadratic forms over integral domains, Acta Arith., 20 (1972), 223251.
[ 3 ] Furuta, Y., A prime decomposition symbol for a non-abelian central extension which is abelian over a bicyclic biquadratic field, Nagoya Math. J., 79 (1980), 79109.
[ 4 ] Furuta, Y., A prime decomposition symbol and integral quadratic forms, Japan. J. Math., 7 (1981), 213216.
[ 5 ] Gauss, C. F., Disquitiones arithmeticae, translation to German by Haser, H., 1889, Chelsea.
[ 6 ] Hooley, C., On the diophantine equation , Arch. Math., 19 (1968), 472478.
[ 7 ] Pall, G., Composition of binary quadratic forms, Bull. Amer. Math. Soc, 54 (1948), 11711175.
[ 8 ] Shanks, D., Gauss’s ternary form reduction and the 2-Sylow subgroup, Math. Comp., 25 (1971), 837853.
[ 9 ] Watson, G. L., Integral quadratic forms, Cambridge Univ. Press, 1960.
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Gauss’s ternary form reduction and its application to a prime decomposition symbol

  • Yoshiomi Furuta (a1)

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