# Questions tagged [ideal-class-group]

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### (Reference request) List of (isomorphism classes of) ideal class groups of $\mathbb{Q}(\sqrt{d})$.

I'm learning about the ideal class group of a number field, and am trying a few exercises where I calculate $\mathbb{Q}(\sqrt{d})$ for $d \in \mathbb{Z}$ for various $d$. I'd like to check my work. ...
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### Calculation of the Picard group and the class group [closed]

I am thinking about the computation of the class group and the Picard group for the case of Number fields over $\mathbb{Q}$ and $\mathbb{F}_p(t)$ Complex varieties I would like to know what kinds of ...
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### There exists imaginary quadratic extension which trivialized 2-part of ideal class group

Let $p$ be a negative prime number such that $p \equiv 5\pmod 8$. Let $K = \mathbb{Q}(\sqrt{p})$ and denote its ideal class group by $Cl_K$. I aim to prove that $Cl_K[2] := \{a \in Cl_K \mid 2a = 0\}$ ...
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### An exact sequence $1 \to {O_{K,S}}^{\times}/{{O_{K,S}}^{\times}}^2 \to K(S,2) \to Cl(K,S)[2] \to 1$

Let $K$ be an imaginary number field.Let $S$ be finite set of places of $K$. Let $K(S,2)\stackrel{\mathrm{def}}{=} \{b\in K^{\times}/{K^{\times}}^2 \mid v(b)≡0\mod2, \forall v\notin S \}$ Let $S-$ ...
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### What is the meaning of the ideal class group?

When I first learned about the ideal class group, I learned that it measures the failure of unique factorization in a number ring. The main justification for this is that a number ring has unique ...
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### Is the class number of $K$ the number of factorizations an element of $\mathcal{O}_K$ can have?

Consider the number field $K = \mathbf{Q}(\sqrt{-5})$, which has ring of integers $\mathcal{O}_K = \mathbf{Z}[\sqrt{-5}]$. It is known that the class number of $K$ is $2$. It is also true that you can ...
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### Finding class number of quadratic number field using Minkowski bound

My understanding of this is as follows: In the general case, one has a quadratic number field $F$, which is always of the form $\mathbb{Q}(\sqrt{d})$ for some square-free integer $d$. Minkowski ...
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