# 二次域和分圆域

Kronecker-Weber 定理还有另一种表述, 即任意拥有交换 Galois 群的代数整数可以表示为单位根的有理数线性和. 比如:
$$\sqrt{5}=e^{2\pi i/5}-e^{4\pi i/5}-e^{6\pi i/5}+e^{8\pi i/5}.$$

Every finite abelian extension of the rational numbers is a subfield of a cyclotomic field. That is, whenever an algebraic number field has a Galois group over the rational numbers that is an abelian group, the field is a subfield of a field obtained by adjoining a root of unity to the rational numbers.

1. 证明 $\mathbb{Q}(\zeta_p)$ 包含 $\sqrt{p}$ 如果 $p\equiv 1\bmod 4$ 或者包含 $\sqrt{-p}$ 如果 $p\equiv 3\bmod 4$. (注: 证明这个结果需要用到关于分圆域 $\mathbb{Q}(\zeta_p)$ 判别式的结论 $$\prod_{1\leq r<s\leq p-1}(\zeta_p^r-\zeta_p^s)^2=\left(\prod_{1\leq r<s\leq p-1}(\zeta_p^r-\zeta_p^s)\right)^2=\pm p^{p-2}.$$ $p\equiv 1\bmod 4$ 时为正, $p\equiv 3\bmod 4$ 时为负. 考虑二次域 $\mathbb{Q}(\sqrt{p})$ 或 $\mathbb{Q}(\sqrt{-p})$ 的判别式.)
2. 证明 $\mathbb{Q}(\zeta_8)$ 包含 $\sqrt{2}$. (注: $e^\frac{2\pi i}{8}+e^\frac{14\pi i}{8}=\sqrt{2}.$)