Nakagawa, Jin Binary forms and orders of algebraic number fields. (English) Zbl 0647.10018 Invent. Math. 97, No. 2, 219-235 (1989). We define a mapping of the space of binary forms of degree \(n>3\) to the space of quadratic forms of n-1 variables. Using the mapping, we obtain a lower estimate for a certain sum of class numbers of totally real binary forms of degree \(n>3\). At the same time, we obtain a lower estimate for the number of orders of totally real algebraic number fields of degree \(n>3\). Further, we prove that there exist infinitely many real quadratic fields having an \(A_ n\)-extension which is unramified at all primes including the infinite primes. Reviewer: J.Nakagawa Cited in 1 ReviewCited in 11 Documents MSC: 11E76 Forms of degree higher than two 11R32 Galois theory 11R80 Totally real fields Keywords:lower estimate; sum of class numbers; totally real binary forms; orders; totally real algebraic number fields; binary form of higher degree; unramified Galois extension PDFBibTeX XMLCite \textit{J. Nakagawa}, Invent. Math. 97, No. 2, 219--235 (1989; Zbl 0647.10018) Full Text: DOI EuDML References: [1] Birch, B.J., Merriman, J.R.: Finiteness theorems for binary forms with given discriminant. Proc. Lond. Math. Soc.24, 385-394 (1972) · Zbl 0248.12002 · doi:10.1112/plms/s3-24.3.385 [2] Cassels, J.W.S.: Rational Quadratic Forms. New York London: Academic Press 1978 · Zbl 0395.10029 [3] Davenport, H.: On the class number of binary cubic forms (I). J. Lond. Math. Soc.26, 183-192 (1951) · Zbl 0044.27002 · doi:10.1112/jlms/s1-26.3.183 [4] Davenport, H., Heilbronn, H.: On the density of discriminants of cubic fields II. Proc. Roy. Soc. Lond. A322, 405-420 (1971) · Zbl 0212.08101 · doi:10.1098/rspa.1971.0075 [5] Lang, S.: Algebraic Number Theory. Reading Mass.: Addison-Wesley 1970 · Zbl 0211.38404 [6] Lang, S.: Fundamentals of Diophantine Geometry, New York Berlin Heidelberg: Springer 1983 · Zbl 0528.14013 [7] Nakagawa, J.: On the Galois group of a number field with square free discriminant. Comment. Math. Univ. Sancti Pauli37, 95-98 (1988) · Zbl 0663.12014 [8] Osada, H.: The Galois groups of the polynomialsX n +aX1+b. J. Number Theory25, 230-238 (1987) · Zbl 0608.12010 · doi:10.1016/0022-314X(87)90029-1 [9] Shintani, T.: On Dirichlet series whose coefficients are class numbers of integral binary cubic forms. J. Math. Soc. Japan24, 132-188 (1972) · Zbl 0227.10031 · doi:10.2969/jmsj/02410132 [10] Uchida, K.: Unramified extensions of quadratic number fields II. Tohoku Math. J.22, 220-224 (1970) · Zbl 0205.35404 · doi:10.2748/tmj/1178242816 [11] Wright, D.J.: The adelic zeta function associated with the space of binary cubic froms I: Global theory. Math. Ann.270, 503-534 (1985) · Zbl 0545.10014 · doi:10.1007/BF01455301 [12] Yamamoto, Y.: On unramified Galois extensions of quadratic number fields. Osaka J. Math.7, 57-76 (1970) · Zbl 0222.12003 [13] Yamamura, K.: On unramified Galois extensions of real quadratic fields. Osaka J. Math.23, 471-478 (1986) · Zbl 0609.12006 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.