# 艾森斯坦整数

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

${\displaystyle z=a+b\omega \,\!}$

${\displaystyle \omega ={\frac {1}{2}}(-1+i{\sqrt {3}})=e^{\frac {2\pi i}{3}}}$

## 性质

${\displaystyle z^{2}-(2a-b)z+(a^{2}-ab+b^{2}).\,\!}$

${\displaystyle {{{{\omega }^{2}}+{\omega }}+{1}}=0}$

${\displaystyle |a+b\omega |^{2}=a^{2}-ab+b^{2}.\,\!}$

${\displaystyle 4a^{2}-4ab+4b^{2}=(2a-b)^{2}+3b^{2},\,\!}$

{±1, ±ω, ±ω2}

## 艾森斯坦素数

xy是艾森斯坦整数，如果存在某个艾森斯坦整数z，使得y = z x，则我们说x能整除y

## 欧几里德域

${\displaystyle N(a+b\,\omega )=a^{2}-ab+b^{2}.\,\!}$

{\displaystyle {\begin{aligned}N(a+b\,\omega )&=|a+b\,\omega |^{2}\\&=(a+b\,\omega )(a+b\,{\bar {\omega }})\\&=a^{2}+ab(\omega +{\bar {\omega }})+b^{2}\\&=a^{2}-ab+b^{2}\end{aligned}}}

## 参考文献

• Bachmann, P. Allgemeine Arithmetik der Zahlkörper. p. 142.
• Cox, D. A. §4A in Primes of the Form x2+ny2: Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, 1989.
• Guy, R. K. "Gaussian Primes. Eisenstein-Jacobi Primes." §A16 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 33-36, 1994.
• Wagon, S. "Eisenstein Primes." §9.8 in Mathematica in Action. New York: W. H. Freeman, pp. 319-323, 1991.