有關二次數體的研究成果相當豐富，其中大部分均可推廣至特徵值為奇質數的二次函數體。至於特徵值為2的二次函數體(被稱為Artin-Schreier擴張)則較為獨特。在本計畫中我們想要探討一些比較特別的定義在多項式環的算術函數以及這些算術函數之間的關係。這些算術函數將有助於計算二次函數體理想類數的平均值問題。二次數體的理想類數的平均值的研究可以追溯到Gauss的時候。對於這個問題，他提出二個猜想，均已被證明。在1990年代，Hoffstein 和 Rosen 考慮特徵值為奇質數的函數體上類推的問題。我們希望考慮特徵值為2時的情況。 There are plenty results about quadratic number fields. Most of them can be generalized to the case of function fields in odd characteristic. For the particular case of characteristic two, it is called an Artin-Schreier extension. In this project, we would like to study some arithmetic functions defined on the polynomial ring over finite fields. Also we would find relations (if exist) between these arithmetic functions. These arithmetic functions play important role in an average value problem. The study of the average value of class numbers of quadratic number fields can be dated back to Gauss. He proposed two conjectures on this problem, both have been proved. In 1990』s, Hoffstein and Rosen considered the analogue of function fields of odd characteristic. We will consider the case of characteristic two. 研究時間：9608 ~ 9709
