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pǿȤ롣ΤȤʬȸƤ֡
1+x+x^2+\cdots+x^{p-1}=0

[]ʬp-1Ĥκ,2,,p-1ʦp=1ˤȤС¤-1Ǥ롣

[]
ǿpˤĤơʬxp-1+xp-2++x+1=0ϡ§黻Ȥ٤ˤäƲ򤱤롣

[]nФʬμϦ(n)Ǥ롣

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ǿpФơΩġ
\gamma~=~\sum_{k=1}^{p-1}~\bigl(~\frac{k}{p}~\bigr)~\omega^k~=~\sqrt{\pm~p}
ǡϡp1ǿʤ+3ǿʤ-롣

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[1]pǿע͡\gamma~=~\sqrt{-p}
[2]pǿǤϤʤע͡\gamma~=~\sqrt{p}

[]p=3ΤȤ13躬x3-1=0򤤤̤Ǥꡢ1,~\omega=\frac{-1+\sqrt{-3}}{2},~\omega=\frac{-1-\sqrt{-3}}{2}Ǥ롣

ǡˡ3˴ؤ1ʿ;2ʿ;Ǥ롣

Τᡢ¤ϼΤ褦˷׻Ǥ롣

\gamma
=~(\frac{1}{3})\omega~+~(\frac{2}{3})\omega^2
=~\omega~-~\omega^2
=~\sqrt{-3}

äơ2\mathbb{Q}(\sqrt{-3})ϱʬQ()ʬΤˤʤ롣

[]ϤΤ褦2ΤʬΤǹͤȤƤΰ̣Τˤʤ뤳Ȥȯ

Ǥϡ̤ͭΤ1Τ٤źäƤǤQ()ȤʬΤΤơʬȸƤǤ롣

[]ͥå
ͭQǤդΥ٥ΤϱʬΤǤ롣

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