臺灣博碩士論文加值系統

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。

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.

1. Dirichlet Characters and L-series 2
1.1. Dirichlet characters 2
1.2. L-series 5
2. Dirichlet’s Theorem 7
2.1. Define ln L(s, χ) 7
2.2. The Key Step 8
2.3. A Proof of Dirichlet’s Theorem 12
3. Quadratic Dirichlet L-series 13
3.1. Summation Formulas of the Series L(1, χ) Associated
to Quadratic Dirichlet Characters 13
3.2. The Character Associated to a Quadratic Field 18
3.3. Some Formulas for the Class Numbers of Quadratic
Fields 28
3.4. Some Examples 34
References 35

[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and New York , PWN-Polish Scientific Publishers , Warszawa , c1990.[5]Steven\ R.\ Finch, Quadratic Dirichlet L-Series, unpublished note, 2005, pp.1--2.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.