hbghlyj
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• $$K$$ a number field, $$\mathcal{O}_K$$ the ring of integers $$=\{a\in\mathbb{Z}:m_{a,\mathbb{Q}}(a)\in\mathbb{Z}(x)\}$$
• Given $$a\in K$$, conjugates of $$a$$ are $$\{\sigma(a):\sigma\in\hom_{\mathbb{Q}}(K,\mathbb{Q})\}$$
• discriminant $$=\det(\sigma_i(\alpha_j))_{i,j}$$ in case of power basis(Vandermonde determinant)$$=(\prod_{1\le i
• Any $$a\unlhd\mathcal{O}_K$$ is $$a=\prod p_i^{e_i}$$ uniquely
• $$Cl(K)=$$ ideals of $$\mathcal{O}_K$$/principal ideals
• How to find $$Cl(K)$$?
• find $$\mathcal{O}_K,\Delta_K,M_K$$ where $$\Delta_K=\begin{cases}4d&otherwise\\d&d\equiv1\pmod4\end{cases}$$ and $$M_K=\begin{cases}\frac2\pi\sqrt{|\Delta_K|}&d<0\\\frac12\sqrt{|\Delta_K|}&d>0\end{cases}$$
• Factorize $$(p)$$ for all $$p\leq M_K$$ to obtain generators
• Find $$x\in\mathcal{O}_K$$ such that $$N(x)$$ is a product of primes $$\le M_K$$ So $$(x)=\prod p_i$$ we get in $$Cl(K)$$: $$1=\prod[p_i]$$
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