[[Imaginary quadratic field]]
# $\mathbb{Q}(\sqrt{ -21 })$

Consider the monogenic [[imaginary quadratic field]] $K = \mathbb{Q}(\alpha)$ where $\alpha = \sqrt{ -21 }$. #m/thm/num/alg 

> [!code]- Sage
> ```sage
> K.<α> = QuadraticField(-21)
> ```

## Discriminant

By [[Discriminant of an algebraic integer]],
$$
\begin{align*}
\Delta_{K} = -84
\end{align*}
$$

## Group of units

By [[Imaginary quadratic field#^P1]],
$$
\begin{align*}
\mathcal{O}_{K}^\times = \{ 1,-1 \}.
\end{align*}
$$

## Class group

[[Minkowski's bound]] is given by
$$
\begin{align*}
M_{K} =  \frac{4\sqrt{ 21 }}{\pi} < 6,
\end{align*}
$$
so applying [[Kummer's factorization theorem]]

| $p$ | $x^2 +21 \bmod p$ | $\langle p \rangle$                  | norms  |
| --- | ----------------- | ------------------------------------ | ------ |
| $2$ | $(x+1)^2$         | $\mathfrak{p}_{2}$                   | $2$    |
| $3$ | $x^2$             | $\mathfrak{p}_{3}^2$                 | $3$    |
| $5$ | $(x+2)(x+3)$      | $\mathfrak{p}_{5} \mathfrak{p}_{5}'$ | $5, 5$ |

Clearly no algebraic integers can have these norms, so we can be satisfied that these are not principal.
Since $\mathfrak{p}_{5}^{-1} \sim \mathfrak{p}_{5}'$, the ideal class group is generated by $\{ [\mathfrak{p}_{1}], [\mathfrak{p}_{2}], [\mathfrak{p}_{3}] \}$.
Some algebraic integers of small [[field norm]] are

| $t$     | $\opn N_{K:\mathbb{Q}}(\alpha+t)$ |
| ------- | --------------------------------- |
| $\pm 1$ | $2 \cdot 11$                      |
| $\pm 2$ | $5^2$                             |
| $\pm 3$ | $2 \cdot 3 \cdot 5$                                  |

whence

- from $t=2$ we see $\mathfrak{p}_{5}^2 = \langle 5,\alpha+2 \rangle^2 = \langle \alpha + 2 \rangle \sim \langle 1 \rangle$;
- from $t=3$ we see $\mathfrak{p}_{2} \mathfrak{p}_{3} \mathfrak{p}_{5} = \langle \alpha-3 \rangle \sim \langle 1 \rangle$.

so we see $\Cl K \cong \mathrm{V}_{4}$.


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