Pagina:Werkecarlf02gausrich.djvu/147

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

Revocatis singulis his modulis ad residua absolute minima secundum modulum ${\textstyle -1+2i}$, animadvertimus, omnes, quibus respondet character ${\textstyle 0}$, esse ${\textstyle \equiv 1}$; eos, quibus character 1 respondet, ${\textstyle \equiv i}$; eos, quorum character est 2, fieri ${\textstyle \equiv -1}$; denique omnes, quorum character est 3, fieri ${\textstyle \equiv -i}$. At characteres numerorum ${\textstyle 1}$, ${\textstyle i}$, ${\textstyle -1}$, ${\textstyle -i}$ pro modulo ${\textstyle -1+2i}$ ipsi sunt ${\textstyle 0}$, ${\textstyle 1}$, ${\textstyle 2}$, ${\textstyle 3}$ resp.; quapropter in omnibus his 17 exemplis character numeri ${\textstyle -1+2i}$ respectu moduli prioris generis ${\textstyle a+bi}$, cum charactere huius numeri respectu moduli ${\textstyle -1+2i}$ identicus est.

Perinde adiumento tabulae invenitur, respondere

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

Revocatis his modulis ad residua minima secundum modulum ${\textstyle -1+2i}$, omnia, quibus resp. characteres ${\textstyle 0}$, ${\textstyle 1}$, ${\textstyle 2}$, ${\textstyle 3}$ respondent, congrua inveniuntur numeris ${\textstyle -1}$, ${\textstyle -i}$, ${\textstyle +1}$, ${\textstyle +i}$; his vero ipsis numeris, si vice versa ${\textstyle -1+2i}$ pro modulo adoptatur, competunt characteres ${\textstyle 2}$, ${\textstyle 3}$, ${\textstyle 0}$, ${\textstyle 1}$ resp. Quapropter in omnibus his 19 exemplis character numeri ${\textstyle -1+2i}$ respectu moduli secundi generis duabus unitatibus differt a charactere huius numeri respectu numeri ${\textstyle -1+2i}$ pro modulo habiti.

Ceterum nullo negotio perspicitur, prorsus similia respectu numeri ${\textstyle -1-2i}$ locum habitura esse.

66.

Pro numero ${\textstyle -3}$ distinctionem inter modulos primi generis et secundi omittimus, quum eventus doceat, illam hic superfluam esse. Respondet itaque