这是一份andrews圣安德鲁斯大学 MT4112作业代写的成功案例
If $S$ is an integral domain, its normalization $\bar{S}$ is the integral closure of $S$ in the quotient field of $S$. An important finiteness result of Emmy Noether tells us that if $S$ is an affine domain, then $\bar{S}$ is a finitely generated $S$-module; in particular, $\bar{S}$ is again an affine domain (see Eisenbud (1995), Corollary 13.13). In other words, if $S$ is of type $S=K\left[x_{1}, \ldots, x_{n}\right] / P$, where $P$ is a prime ideal, then $\bar{S}$ is of type $K\left[y_{1}, \ldots, y_{m}\right] / P^{\prime}$, where $P^{\prime}$ is a prime ideal. To compute the normalization means to find such a representation for $\bar{S}$ together with the normalization map
$$
S=K\left[x_{1}, \ldots, x_{n}\right] / P \hookrightarrow \bar{S}=K\left[y_{1}, \ldots, y_{m}\right] / P^{\prime}
$$
More generally, if $S$ is any reduced ring, its normalization $\bar{S}$ is the integral closure of $S$ in the total quotient ring of $S$. In conjunction with Noether’s finiteness result, the theorem on the splitting of normalization tells us that if $S$ is a reduced affine ring, then $\bar{S}$ may be written as a product of affine domains. More precisely, if $S$ is of type $S=K\left[x_{1}, \ldots, x_{n}\right] / I$, where $I$ is a radical ideal, and if $P_{1}, \ldots, P_{s}$ are the (minimal) associated primes of $I$, then
$$
\bar{S} \cong \overline{\left(K\left[x_{1}, \ldots, x_{n}\right] / P_{1}\right)} \times \cdots \times \overline{\left(K\left[x_{1}, \ldots, x_{n}\right] / P_{s}\right)}
$$
MT4112 COURSE NOTES ：
The degrees $d_{j}$ of the secondary invariants and their number $\mu_{j}$ in each degree are uniquely determined as the exponents and the coefficients of the polynomial
$$
\sum_{j=0}^{b} \mu_{j} t^{d_{j}}:=H_{K[x]^{c}}(t) \cdot \prod_{i=1}^{n}\left(1-t^{\operatorname{deg}\left(p_{i}\right)}\right)
$$
The total number of secondary invariants is
$$
\frac{1}{|G|} \prod_{i=1}^{n} \operatorname{deg}\left(p_{i}\right)
$$