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hbghlyj
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I'm learning by reading math and asking questions

  • $K$ 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)$=(\prod_{1\le i<j\le n}(\alpha_i-\alpha_j))^2$
  • 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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