在數學的領域中，屬於代數的這部分給人一種他就是要計算一些簡單或是複雜的式子。
好比說去計算一個集合中所含元素的個數；或是我們給出一個複雜多樣的集合後，有沒有辦法把他一步一步地分解
，轉換成我們所熟悉的基本結構，像群，環，體之類的推廣。而計算維度或是長度的話，
是可以幫助我們去瞭解彼此之間的關係是怎樣的形式。也就是說；我們能不能利用基本結構來描述任意給出
的集合呢？近幾十年來，一直都有人在研究相關的課題。而 Paul Monsky從1963年開始，也陸續在一些有名的
雜誌，如 Mathematische Annalen，
Mathematische Zeitschrift，... 等，發表一些他在數論上的工作(像 Class numbers，
Class groups 的探討)。
這幾年他開始去計算一些特殊環的 Hilbert-Kunz functions，主要是以數論的手法作為工具。例如他在計算特徵數為$2$的三次型的 Hilbert-Kunz functions，用到了 elliptic curves,$j$-invariants。
因此我選擇這個主題作為我的論文主題。本篇論文共分為三章，本章做簡單的介紹，第二章所有需要的符，
定義和基本性質，而本篇最重要的定理我把它擺在第三章，最後的部分我將所需要的參考資料詳細地列出來。

[1] R. O. Buchweitz and Q. Chen, Hilbert-Kunz functions of cubic
curves and surfaces. J.Algebra 197 (1997), 246-267.
[2] Li,Chiang ; Yu Ching, Hung , On Hilbert-Kunz function and
representation ring. Bull.Inst. Math. Acad. Sinica 26 (1998), no. 1, 1-32.
[3] Li,Chiang ; Yu Ching, Hung , On Hilbert-kunz functions of
some hypersurfaces. J. Algebra 199 (1998), no. 2, 499-527.
[4] Aldo, Conca, Hilbert-Kunz function of monomial ideals and binomial
hypersurfaces. Manu. Math. 90 (1996), 287--300.
[5] M. Contessa, On the Hilbert-Kunz function and Koszul homology.
J. Algebra 175 (1995), 757-766.
[6] Lawden, Derek F, Elliptic functions and Applications.
[7] C. Han and P. Monsky, Some surprising Hilbert-Kunz functions.
Math. Z. 214 (1993), 119-135.
[8] Dale Husemoller, Elliptic Curves. Springer-Verlag New York. GTM 111 (1987)
[9] Anthony W. Knapp, Ellliptic curves. Princenton University
Press. Mathematical Notes. 40.
[10] Koblitz, Introduction to Elliptic Curives and Modular.
Springer-Verlag New York. GTM 97 (1984)
[11] Ernst, Kunz, On noetherian rings of characteristic p. Amer. J. Math.
98 (1976), 999-1013. [12] Ernst, Kunz, Remark on the purity of the branch locus. Proceedings of the
American Math. Society. 20 (1969), 378-380.
[13] Ernst, Kunz, Characterizations of regular local rings of characteristic p.
Amer. J. Math. 91 (1969),772-784. [14] Hideyuki, Matsumura; P. Monsky, On the automorphisms of
hypersurfaces. J. Math. Kyoto Univ. 3 1963/1964 347-361.
[15] P. Monsky, Hilbert-Kunz functions in a family: line-
S(4)quartics. J.Alebra 208 (1998), no. 1, 359-371.
[16] P. Monsky, Hilbert-Kunz functions in a family: point-
S(4)quartics. J.Alebra 208 (1998), no. 1, 343-358.
[17] P. Monsky, The Hilbert-Kunz function of a characteristic
2 cubic. J.Algebra 197 (1997), no. 1, 268-277.
[18] P. Monsky, Calculating a trapezoidal spectrum. Amer.Math.
Monthly 103 (1996), no. 6, 500-501.
[19] P. Monsky, Simplifying the proof of Dirichlet`s theorem.
Amer. Math.Monthly 100 (1993), no. 9, 861-862.
[20] P. Monsky, "Three constructions of rational points on
Y(2)=X(3)+NX"[Math. Z. 209 (1992), no. 3, 445-462]
Math. Z. 212 (1993), no. 1, 141.
[21] P. Monsky, Three constructions of rational points on Y(2)=X(3)+NX Math.Z. 209 (1992), no. 3, 445-462.
[22] P. Monsky, A conjecture of Stein on plane dissections.
Math. Z. 205 (1990), no. 4, 583-592.
[23] P. Monsky, Mock Heener points ad congruent numbers.
Math. Z.204 (1990), no. 1, 45-67.
[24] P. Monsky, Class numbers in Z(d)(p)-extensions. IV.
Math. Z. 196 (1987), no. 4, 547-572. [25] P. Monsky, Class numbers in Z(d)(p)-extensions.
III. Math. Z. 193 (1986), no. 4, 491-514.
[26] P. Mnosky, Class numbers in Z(d)(p)-extensions.
II. Math. Z. 191 (1986), no. 3, 377-395.
[27] P. Monsky,p-ranks of class groups in Z(p)-
extensions.Math. Ann. 263 (1983), no. 4, 509-514.
[28] P. Monsky, On p-adic power series. Math. Ann.
255 (1981), no. 2, 217-227.
[29] P. Monsky, Class numbers in Z(d)(p)-extensions.
Math. Ann. 255 (1981), no. 2, 235-258.
[30] P. Monsky, Some invariants of Z(d)(p)-extensions.
Math. Ann. 255 (1981), no. 2, 229-233.
[31] P. Mnodky, Formal cohomology. III. Fixed point theorems.
Ann. of Math.2 93 (1971), 315-343.
[32] P. Monsky, p-adic analysis and zata functions. Lectures
in Mathematics, Department of Mathematics, Kyoto University, 4 Kinokuniya
Book-Store Co., Ltd., Tokyo (1970) iv+117 pp.
[33] P. Monsky, On dividing a square into triangles. Amer.
Math. Monthly 77 (1970) 161-164.
[34] Lang, Serge, Elliptic functions. Addison-Wesley Publishing Company. (1973)
[35] Lang, Serge, Algebra Third Edition. Addison-Wesley Publishing Company. (1927)
[36] Larry, Smith, Linear Algebra Second Edition. Springer-Verlag New York. UTM (1984)