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本帖最后由 hbghlyj 于 2024-2-21 21:06 编辑 2007年Q2是
If $\mathcal{O}$ is a Dedekind domain, show that any ideal $I$ can be generated by at most two elements.
Hint: use the preceding exercise.
将$I$分解为$\mathfrak{p}_1^{e_1}\cdots\mathfrak{p}_n^{e_n}$。
任取$\alpha\in I-\{0\}$，
由于$I\mid (\alpha)$，$(\alpha)$分解为$\mathfrak{p}_1^{f_1}\cdots\mathfrak{p}_n^{f_n}\mathfrak{q}_1^{g_1}\cdots\mathfrak{q}_m^{g_m}\;(f_i\geqslant e_i)$。
(CRT)存在$\beta\in\mathcal{O}$，使得$v_{\mathfrak{p}_i}(\beta)=e_i$且$v_{\mathfrak{q}_i}(\beta)=0$。
$$(\alpha,\beta)=\gcd((\alpha),(\beta))=\mathfrak{p}_1^{e_1}\cdots\mathfrak{p}_n^{e_n}=I$$
这表明对于任何$I$和任何$\alpha\in I-\{0\}$总可以找到一个互补生成元。
2024年Q2是（第1小问是上述使用的 CRT，第2小问与上述相同）
Let $K$ be a number field, and let $I, J \subset \mathcal{O}_K$ be non-zero ideals.
(1) Determine the factorisations into prime ideals of $I+J$ and $I \cap J$ in terms of those for $I$ and $J$. Show that if $I+J=\mathcal{O}_K$ then $I \cap J=I J$ and there is an isomorphism of rings $\mathcal{O}_K / I J \cong \mathcal{O}_K / I \times \mathcal{O}_K / J$.
(2) Show that $I$ can be generated by at most 2 elements.
(3) Let $\phi(I)=\left|\left(\mathcal{O}_K / I\right)^{\times}\right|$. Show that
\[
\phi(I)=\mathrm{N}(I) \prod_{P \mid I}\left(1-\frac{1}{\mathrm{N}(P)}\right),
\]
where the product is over the set of prime ideals $P$ dividing $I$. |
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