Pagina:Werkecarlf02gausrich.djvu/67

ubi signum superius vel inferius valebit, prout ${\textstyle p}$ est formae ${\textstyle 4k+1}$ vel formae ${\textstyle 4k+3}$. Et quum insuper ${\textstyle f(x^{\alpha ^{{\frac {1}{2}}(p-1)}+1})-p}$ divisibilis sit per ${\textstyle {\frac {1-x^{p}}{1-x}}}$, erit etiam ${\textstyle \xi \xi \mp p}$ per ${\textstyle {\frac {1-x^{p}}{1-x}}}$ divisibilis. Q. E. D.

Ne duplex signum ullam ambiguitatem adducere possit, per ${\textstyle \varepsilon }$ numerum ${\textstyle +1}$ vel ${\textstyle -1}$ denotabimus, prout ${\textstyle p}$ est formae ${\textstyle 4k+1}$ vel ${\textstyle 4k+3}$. Erit itaque ${\textstyle {\frac {(1-x)(\xi \xi -\varepsilon p)}{1-x^{p}}}}$ functio integra ipsius ${\textstyle x}$, quam per ${\textstyle Z}$ designabimus.

4.

Sit ${\textstyle q}$ numerus positivus impar, adeoque ${\textstyle {\frac {1}{2}}(q-1)}$ integer. Erit itaque ${\textstyle (\xi \xi )^{{\frac {1}{2}}(q-1)}-(\varepsilon p)^{{\frac {1}{2}}(q-1)}}$ divisibilis per ${\textstyle \xi \xi -\varepsilon p}$, et proin etiam per ${\textstyle {\frac {1-x^{p}}{1-x}}}$. Statuamus ${\textstyle \varepsilon ^{{\frac {1}{2}}(q-1)}=\delta }$, atque

$$\xi ^{q-1}-\delta p^{{\frac {1}{2}}(q-1)}={\frac {1-x^{p}}{1-x}}\cdot Y$$

eritque ${\textstyle Y}$ functio integra ipsius ${\textstyle x}$, atque ${\textstyle \delta =+1}$, quoties unus numerorum ${\textstyle p}$, ${\textstyle q}$. sive etiam uterque, est formae ${\textstyle 4k+1}$; contra erit ${\textstyle \delta =-1}$, quoties uterque ${\textstyle p}$, ${\textstyle q}$ est formae ${\textstyle 4k+3}$.

5.

Iam supponamus, ${\textstyle q}$ quoque esse numerum primum (a ${\textstyle p}$ diversum) patetque per theorema in Disquisitionibus Arithmeticis art. 51 demonstratum,

$$\xi ^{q}-(x^{q}-x^{q\alpha }+x^{q\alpha \alpha }-x^{q\alpha ^{3}}+{\text{etc.}}-x^{q\alpha ^{p-2}})$$

divisibilem fieri per ${\textstyle q}$, sive formae ${\textstyle qX}$, ita ut ${\textstyle X}$ sit functio integra ipsius ${\textstyle x}$ etiam respectu coëfficientium numericorum (quod etiam de functionibus reliquis integris hic occurrentibus ${\textstyle Z}$, ${\textstyle Y}$, ${\textstyle W}$ subintelligendum est). Designemus pro modulo ${\textstyle p}$ atque radice primitiva ${\textstyle \alpha }$ indicem numeri ${\textstyle q}$ per ${\textstyle \mu }$, i.e. sit ${\textstyle q\equiv \alpha ^{\mu }{\pmod {p}}}$. Erunt itaque numeri ${\textstyle q}$, ${\textstyle q\alpha }$, ${\textstyle q\alpha \alpha }$, ${\textstyle q\alpha ^{3}\ldots q\alpha ^{p-2}}$ secundum modulum ${\textstyle p}$ resp. congrui numeris ${\textstyle \alpha ^{\mu }}$, ${\textstyle \alpha ^{\mu +1}}$, ${\textstyle \alpha ^{\mu +2}\ldots \alpha ^{p-2}}$, ${\textstyle 1}$, ${\textstyle \alpha }$, ${\textstyle \alpha \alpha \ldots \alpha ^{\mu -1}}$, adeoque

$${\begin{aligned}&x^{q}-x^{\alpha ^{\mu }}\\&x^{q\alpha }-x^{\alpha ^{\mu +1}}\\&x^{q\alpha \alpha }-x^{\alpha ^{\mu +2}}\\&x^{q\alpha ^{3}}-x^{\alpha ^{\mu +3}}\\&\quad \vdots \end{aligned}}$$