朱樺臺灣大學：數學研究所徐祥峻Hsu, Hsiang-ChunHsiang-ChunHsu2007-11-282018-06-282007-11-282018-06-282007http://ntur.lib.ntu.edu.tw//handle/246246/59400令 $Bbb F_q$ 是有 $q$ 個元素的 Galois 體, $Q_n$ 是 $Bbb F_q^n$ 上的非退化二次型且 $O_n(Bbb F_q)$ 是由 $Q_n$ 定義的正交群。 令 $O_n(Bbb F_q)$ 線性地作用於多項式環 $Bbb F_q[x_1,x_2,dots,x_n]$ 上。 在本論文中, 我們將確切地 找出 $O_n(Bbb F_q)$ 的不變子環的生成元及其關係, 並且證明此不變子環是唯一分解環及完全交。Let $Bbb F_q$ be the Galois field with $q$ elements, $Q_n$ a non-degenerated quadratic form on $Bbb F_q^n$, and $O_n(Bbb F_q)$ the orthogonal group defined by $Q_n$. Let $O_n(Bbb F_q)$ act linearly on the polynomial ring $Bbb F_q[x_1,x_2,dots,x_n]$. In this paper, we will find explicit generators and relations for the ring of invariants of $O_n(Bbb F_q)$, and prove that it is a UFD and a complete intersection.Acknowledgements..........................................i Abstract in Chinese......................................ii Abstract................................................iii Contents.................................................iv 1 Introduction............................................1 2 The Notations and Elementary Properties.................5 3 The Statements of The Main Theorems....................20 4 Lemmas.................................................25 4.1 Two Key Lemmas.....................................25 4.2 Formulae on Polynomials Modulo Variables...........30 4.3 A Lemma on $R_n^{pm}$..............................38 4.4 Properties of Regular Sequence.....................43 5 Proposition $O_k$......................................48 6 The Proof of Theorem $O_{k+1}$.........................61 7 The Proof of Theorem $E_k^+$...........................71 8 The Proof of Theorem $E_k^-$...........................73 Reference................................................80475067 bytesapplication/pdfen-US模正交群不變多項式完全交modular orthogonal grouppolynomial invariantscomplete intersectionthesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/59400/1/ntu-96-R94221009-1.pdf