- 學位論文
二次體中的 class number 與 Dirichlet 的 L 級數
Class Numbers of Quadratic Fields and Dirichlet L-series
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摘要
在本論文中，我們完整的證明了Dirichlet’s theorem。另外，我們針對在Quadratic fields中的L-series作研究，寫成許多不同的形式。利用L-series在s=1時的不同形式可推導出class number的一些公式。由class number最後推導出的公式可看出quadratic residues和quadratic nonresidues的個數在區間 (0,p/2) 的分布情形。最後，我們給了一些在quadratic fields中，L-series在s=1時，用不同的character代入所得到的特殊值。
並列摘要
In this thesis, we first review some well-known results about the Dirichlet characters and L-series. Then we give an elaborate proof of Dirichlet's theorem. Besides, we are interesting in the formulas of L-series associated to the quadratic fields. Then we compute class numbers of quadratic fields with L-series at $s=1$ and the Gaussian sum. In the end, examples of L-series at $s=1$ are given.
參考文獻
[1]Kenneth F. Ireland, Michael I. Rosen, A Classical Introduction to Modern Number Theory, Springer.
[2]Lawrence C. Washington, Introduction to Cyclotomic Fields, Springer, New York, c1982.
[3]Z. I. Borevich, I. R. Shafarevich, Number Theory, Academic Press, New York, 1966.
[4]}Wladyslaw Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Springer-Verlag, Berlin and New York , PWN-Polish Scientific Publishers , Warszawa , c1990.[5]Steven R. Finch, Quadratic Dirichlet L-Series, unpublished note, 2005, pp.1--2.
[6]James K. Strayer, Elemetary Number Theory, PWS Pub. Co., Boston, c1994.