Abstract
There are close relations between Bernoulli numbers and quadratic fields or quadratic forms. In order to explain those, we give a survey of the ideal theory of quadratic fields and quadratic forms. Since Gauss , it is well known that there is a deep relation between the ideal theory of quadratic fields (i.e. quadratic extensions of the rational number field) and integral quadratic forms. This is obvious for specialists, but textbooks which explain this in detail are rare. In most textbooks, they treat the correspondence of ideals only in the case of maximal orders (the ring of all integers) for which the description is simple and easy. This is very disappointing. We cannot see the whole picture of the theory of quadratic forms by such a restricted treatment and sometimes it causes misunderstanding. We need the full description of this kind of relation when we explain later the relation between L-functions of prehomogeneous vector spaces and the Bernoulli numbers. So it would be a good chance to try to explain the full relation here.
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Notes
- 1.
Here half-integers mean their doubles are integers.
- 2.
In Section 304 of his book on number theory, Disquisitiones Arithmeticae [35], Gauss stated his speculation on this and also his reasonable insight on the relation to the size of fundamental units .
References
Dirichlet, P.G.L.: Lectures on Number Theory, with Supplements by R. Dedekind. History of Mathematics Sources, vol. 16. Amer. Math. Soc. (1999)
Gauss, C.F.: Disquisitiones Arithmeticae (1801)
Serre, J.-P.: Cours d’arithmétique, Presses Universitaires de France, 1970. English translation: A course in arithmetic, Graduate Text in Mathematics, vol. 7. Springer (1973)
Zagier, D.: Zetafunktionen und Quadratische Körper. Springer (1981)
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Ibukiyama, T., Kaneko, M. (2014). Quadratic Forms and Ideal Theory of Quadratic Fields. In: Bernoulli Numbers and Zeta Functions. Springer Monographs in Mathematics. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54919-2_6
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DOI: https://doi.org/10.1007/978-4-431-54919-2_6
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