Pagina:Werkecarlf02gausrich.djvu/145

$${\begin{array}{c|l}{\text{characterem }}&{\text{ modulis primi generis.}}\\\hline 0&{\phantom {+}}5+4i,-7+8i,-7-8i,-11+4i\\1&{\phantom {+}}1-4i,-3+8i,-3-8i,9+4i,-11\\2&{\phantom {+}}5-4i,-7,-11-4i\\3&-3,1+4i,5+8i,5-8i,9-4i\\\end{array}}$$

Si haec septemdecim exempla attente consideramus, in omnibus invenimus characterem ${\textstyle \equiv {\frac {1}{4}}(a-b-1){\pmod {4}}}$.

Perinde respondet

$${\begin{array}{c|l}{\text{character}}&{\text{modulis secundi generis.}}\\\hline 0&{\phantom {+}}3-2i,-1-6i,7+2i,-5+6i,-1+10i,11+6i\\1&-5+2i,-1+6i,7-2i,-1-10i,3+10i\\2&-1+2i,-5-2i,3-10i,7+10i\\3&-1-2i,3+2i,-5-6i,7-10i,11-6i\\\end{array}}$$

In omnibus his viginti exemplis, levi attentione adhibita, invenitur character ${\textstyle \equiv {\frac {1}{4}}(a-b-5){\pmod {4}}}$.

Facile has duas regulas in unam pro utroque modulorum genere valentem contrahere licet, si perpendimus, ${\textstyle {\frac {1}{4}}bb}$ esse pro modulis prioris generis ${\textstyle \equiv 0}$, pro modulis posterioris generis ${\textstyle \equiv 1{\pmod {4}}}$. Est itaque character numeri ${\textstyle 1+i}$ respectu moduli cuiusvis primi inter associatos primarii ${\textstyle \equiv {\frac {1}{4}}(a-b-1-bb){\pmod {4}}}$.

Obiter hic annotare convenit, quum ${\textstyle (b+1)^{2}}$ semper sit formae ${\textstyle 8n+1}$, sive ${\textstyle {\frac {1}{4}}(2b+bb)}$ par, characterem istum semper parem vel imparem fieri, prout ${\textstyle {\frac {1}{4}}(a+b-1)}$ par sit vel impar, quod quadrat cum regula pro charactere quadratico in art. 58 prolata.

Quum ${\textstyle {\frac {1}{4}}(a-b-1)}$, ${\textstyle {\frac {1}{4}}(a-b+3)}$ sint integri, quorum alter par, alter impar, ipsorum productum par erit, sive ${\textstyle {\frac {1}{8}}(a-b-1)(a-b+3)\equiv 0{\pmod {4}}}$. Hinc loco expressionis allatae pro charactere biquadratico haec quoque adoptari potest

$${\frac {1}{4}}(a-b-1-bb)-{\frac {1}{8}}(a-b-1)(a-b+3)={\frac {1}{8}}(-aa+2ab-3bb+1)$$

quae forma eo quoque nomine se commendat, quod non restringitur ad modulos primarios, sed tantummodo supponit; ${\textstyle a}$ esse imparem, ${\textstyle b}$ parem: manifesto enim in hac suppositione vel ${\textstyle a+bi}$, vel ${\textstyle -a-bi}$ erit numerus inter associatos primarius, valorque istius formulae pro utroque modulo idem.