# 高斯整數

${\displaystyle \mathbb {N} \subseteq \mathbb {Z} \subseteq \mathbb {Q} \subseteq \mathbb {R} \subseteq \mathbb {C} }$

${\displaystyle \mathbf {Z} [i]=\{a+bi\mid a,b\in \mathbb {Z} \}}$

${\displaystyle N(zw)=N(z)N(w)}$

${\displaystyle \mathbf {Z} [i]}$單位元${\displaystyle 1,-1,i,-i}$的範數均為${\displaystyle 1}$

## 高斯整環

### 質元素

${\displaystyle \mathbf {Z} [i]}$ 素元素又称为高斯質數

• ${\displaystyle a,b}$ 中有一个是零，另一个是形为${\displaystyle 4n+3}$ 或其相反数${\displaystyle -(4n+3)}$ 的素数

• ${\displaystyle a,b}$ 均不为零，而${\displaystyle a^{2}+b^{2}}$ 为素数。

### 作为欧几里德环

${\displaystyle \mathbb {Z} [i]/{\left\langle a+bi\right\rangle }\cong \mathbb {Z} _{a^{2}+b^{2}}\ =\ \{[0],[1],[2]\cdots ,[a^{2}+b^{2}-1]\}.}$ [1]

## 参考文献

1. ^ 存档副本. [2022-01-01]. （原始内容存档于2015-09-23）.
• C. F. Gauss, Theoria residuorum biquadraticorum. Commentatio secunda., Comm. Soc. Reg. Sci. Gottingen 7 (1832) 1-­34; reprinted in Werke, Georg Olms Verlag, Hildesheim, 1973, pp. 93-­148.
• 从数到环：环论的早期历史，由Israel Kleiner所作 (Elem. Math. 53 (1998) 18 – 35)
• Ribenboim, Paulo, The New Book of Prime Number Records, New York: Springer, 1996, ISBN 0-387-94457-5