這篇論文主要的結果是希望具體描述“正交群作用在佈於有限體的多項式環上其不變子環的生成元”。首先我們在不變子環裡找出若干元素，去證明不變子環is integral over這些元素所生成的子環(我們記為R_{n}^{*})，並且證明了不變子環與R_{n}^{*}有相同的商體；最後我們證明了當n=2,4時，R_{n}^{*}是一個UFD，因此R_{n}^{*} is integrally closed，所以我們可以得知不變子環在n=2,4時的生成元。所謂正交群是指保持二次型之所有可逆線性變換中所構成的乘法群；在此篇論文中，我們特別強調此二次型為Q_{n}^{-}=x_{1}^{2}-x_{2}^{2}+...+x_{n-1}^{2}-εx_{n}^{2}，其中n是偶數且ε是一個在有限體裡的非平方數。

Let Q_{n}^{-}=x_{1}^{2}-x_{2}^{2}+...+x_{n-1}^{2}-εx_{n}^{2}
be a nondegenerate quadratic form over the
finite field Fq with charFq is not equal to 2 where ε is a non-square in Fq, and let O(Fq^{n},Q_{n}^{-}) be the associated orthogonal group. Let O(Fq^{n},Q_{n}^{-}) act linearly on the polynomial ring
Fq[x_{1},...,x_{n}]. In this thesis we try to find the invariant subring Fq[x_{1},...,x_{n}]^{O(Fq^{n},Q_{n}^{-})}
with explicit generators. In fact, we find the invariant elements in Fq[x_{1},...,x_{n}]^{O(Fq^{n},Q_{n}^{-})},
and denote the subring which these invariant elements generate by R_{n}^{*}. We show that
Fq[x_{1},...,x_{n}]^{O(Fq^{n},Q_{n}^{-})} is integral over R_{n}^{*}, and show that Fq[x_{1},...,x_{n}]^{O(Fq^{n},Q_{n}^{-})} and R_{n}^{*} have the same quotient field. As n=2,4, we show that R_{n}^{*} is a UFD, and hence is
integrally closed. Thus Fq[x1,x2]^{O(Fq^{2},Q_{2}^{-})}=R_{2}^{*} and Fq[x1,x2,x3,x4]]^{O(Fq^{4},Q_{4}^{-})}=R_{4}^{*}, so we get the generators of invariant subring when n=2,4.

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