Pagina:Werkecarlf02gausrich.djvu/152

dum modulum ${\textstyle m}$ inter complexus ${\textstyle C}$, ${\textstyle C^{\prime }}$, ${\textstyle C^{\prime \prime }}$, ${\textstyle C^{\prime \prime \prime }}$ distributis, multitudo eorum, quae ad singulos hos complexus pertinent, resp. per ${\textstyle c}$, ${\textstyle c^{\prime }}$, ${\textstyle c^{\prime \prime }}$, ${\textstyle c^{\prime \prime \prime }}$ denotatur: character numeri ${\textstyle k}$ respectu moduli ${\textstyle m}$ erit ${\textstyle \equiv c^{\prime }+2c^{\prime \prime }+3c^{\prime \prime \prime }{\pmod {4}}}$.

Demonstr. Sint illa ${\textstyle c}$ residua minima ad ${\textstyle C}$ pertinentia ${\textstyle \alpha }$, ${\textstyle \beta }$, ${\textstyle \gamma }$, ${\textstyle \delta }$ etc.; dein ${\textstyle c^{\prime }}$ residua ad ${\textstyle C^{\prime }}$ pertinentia haec ${\textstyle m+i\alpha ^{\prime }}$, ${\textstyle m+i\beta ^{\prime }}$, ${\textstyle m+i\gamma ^{\prime }}$, ${\textstyle m+i\delta ^{\prime }}$ etc.; porro ${\textstyle c^{\prime \prime }}$ residua ad ${\textstyle C^{\prime \prime }}$ pertinentia haec ${\textstyle (1+i)m-\alpha ^{\prime \prime }}$, ${\textstyle (1+i)m-\beta ^{\prime \prime }}$, ${\textstyle (1+i)m-\gamma ^{\prime \prime }}$, ${\textstyle (1+i)m-\delta ^{\prime \prime }}$ etc.; denique ${\textstyle c^{\prime \prime \prime }}$ residua ad ${\textstyle C^{\prime \prime \prime }}$ pertinentia haec ${\textstyle im-i\alpha ^{\prime \prime \prime }}$, ${\textstyle im-i\beta ^{\prime \prime \prime }}$, ${\textstyle im-i\gamma ^{\prime \prime \prime }}$, ${\textstyle im-i\delta ^{\prime \prime \prime }}$ etc. Iam consideremus quatuor producta, scilicet

1. productum ex omnibus ${\textstyle {\frac {1}{4}}(p-1)}$ numeris complexum ${\textstyle C}$ constituentibus:
2. productum productorum, quae e multiplicatione singulorum horum numerorum per ${\textstyle k}$ orta erant;
3. productum e residuis minimis horum productorum, puta e numeris ${\textstyle \alpha }$, ${\textstyle \beta }$, ${\textstyle \gamma }$, ${\textstyle \delta }$ etc., ${\textstyle m+i\alpha ^{\prime }}$, ${\textstyle m+i\beta ^{\prime }}$ etc. etc.
4. productum ex omnibus ${\textstyle c+c^{\prime }+c^{\prime \prime }+c^{\prime \prime \prime }}$ numeris ${\textstyle \alpha }$, ${\textstyle \beta }$, ${\textstyle \gamma }$, ${\textstyle \delta }$ etc., ${\textstyle \alpha ^{\prime }}$, ${\textstyle \beta ^{\prime }}$, ${\textstyle \gamma ^{\prime }}$, ${\textstyle \delta ^{\prime }}$ etc., ${\textstyle \alpha ^{\prime \prime }}$, ${\textstyle \beta ^{\prime \prime }}$, ${\textstyle \gamma ^{\prime \prime }}$, ${\textstyle \delta ^{\prime \prime }}$ etc., ${\textstyle \alpha ^{\prime \prime \prime }}$, ${\textstyle \beta ^{\prime \prime \prime }}$, ${\textstyle \gamma ^{\prime \prime \prime }}$, ${\textstyle \delta ^{\prime \prime \prime }}$ etc.

Denotando haec quatuor producta ordine suo per ${\textstyle P}$, ${\textstyle P^{\prime }}$, ${\textstyle P^{\prime \prime }}$, ${\textstyle P^{\prime \prime \prime }}$, manifesto erit

$$P^{\prime }=k^{{\frac {1}{4}}(p-1)}P,\quad P^{\prime }\equiv P^{\prime \prime },\quad P^{\prime \prime }\equiv P^{\prime \prime \prime }i^{c^{\prime }+2c^{\prime \prime }+3c^{\prime \prime \prime }}{\pmod {m}}$$

et proin

$$Pk^{{\frac {1}{4}}(p-1)}\equiv P^{\prime \prime \prime }i^{c^{\prime }+2c^{\prime \prime }+3c^{\prime \prime }}{\pmod {m}}$$

At facile perspicietur, numeros ${\textstyle \alpha ^{\prime }}$, ${\textstyle \beta ^{\prime }}$, ${\textstyle \gamma ^{\prime }}$, ${\textstyle \delta ^{\prime }}$ etc., ${\textstyle \alpha ^{\prime \prime }}$, ${\textstyle \beta ^{\prime \prime }}$, ${\textstyle \gamma ^{\prime \prime }}$, ${\textstyle \delta ^{\prime \prime }}$ etc., ${\textstyle \alpha ^{\prime \prime \prime }}$, ${\textstyle \beta ^{\prime \prime \prime }}$, ${\textstyle \gamma ^{\prime \prime \prime }}$, ${\textstyle \delta ^{\prime \prime \prime }}$ etc. omnes ad complexum ${\textstyle C}$ pertinere, atque tum inter se tum a numeris ${\textstyle \alpha }$, ${\textstyle \beta }$, ${\textstyle \gamma }$, ${\textstyle \delta }$ etc. diversos esse, sicuti hi ipsi inter se diversi sint. Omnes itaque hi numeri simul sumti, et abstrahendo ab ordine, prorsus identici esse debent cum omnibus numeris complexum ${\textstyle C}$ constituentibus, unde colligimus ${\textstyle P=P^{\prime \prime \prime }}$, adeoque

$$Pk^{{\frac {1}{4}}(p-1)}\equiv Pi^{c^{\prime }+2c^{\prime \prime }+3c^{\prime \prime \prime }}{\pmod {m}}$$

Denique quum singuli factores producti ${\textstyle P}$ per ${\textstyle m}$ non sint divisibiles, hinc concluditur

$$k^{{\frac {1}{4}}(p-1)}\equiv i^{c^{\prime }+2c^{\prime \prime }+3c^{\prime \prime \prime }}{\pmod {m}}$$

unde ${\textstyle c^{\prime }+2c^{\prime \prime }+3c^{\prime \prime \prime }}$ erit character numeri ${\textstyle k}$ respectu moduli ${\textstyle m}$. Q. E. D.