Pagina:Werkecarlf02gausrich.djvu/82

$${\begin{array}{c|l}{\text{ad}}&{\text{residua minima numerorum}}\\\hline A&1,{\phantom {g}}g^{4},g^{8},{\phantom {g}}g^{12}\ldots \ldots g^{p-5}\\B&g,{\phantom {g}}g^{5},g^{9},{\phantom {g}}g^{13}\ldots \ldots g^{p-4}\\C&gg,g^{6},g^{10},g^{14}\ldots \ldots g^{p-3}\\D&g^{3},g^{7},g^{11},g^{15}\ldots \ldots g^{p-2}\end{array}}$$

Hinc omnes propositiones praecedentes sponte demanant.

Ceterum sicuti hic numeri ${\textstyle 1}$, ${\textstyle 2}$, ${\textstyle 3\ldots p-1}$ in quatuor classes distributi sunt, quarum complexus per ${\textstyle A}$, ${\textstyle B}$, ${\textstyle C}$, ${\textstyle D}$ designamus, ita quemvis integrum per ${\textstyle p}$ non divisibilem, ad normam ipsius residui minimi secundum modulum ${\textstyle p}$, alicui harum classium adnumerare licebit.

9.

Denotabimus per ${\textstyle f}$ residuum minimum potestatis ${\textstyle g^{{\frac {1}{4}}(p-1)}}$ secundum modulum ${\textstyle p}$, unde quum fiat ${\textstyle ff\equiv g^{{\frac {1}{2}}(p-1)}\equiv -1}$ (Disquis. Arithm. art. 62), patet, characterem ${\textstyle f}$ hic idem significare quod in art. 6. Potestas ${\textstyle g^{{\frac {1}{4}}\lambda (p-1)}}$ itaque, denotante ${\textstyle \lambda }$ integrum positivum, congrua erit secundum modulum ${\textstyle p}$ numero ${\textstyle 1}$, ${\textstyle f}$, ${\textstyle -1}$, ${\textstyle -f}$, prout ${\textstyle \lambda }$ formae ${\textstyle 4m}$, ${\textstyle 4m+1}$, ${\textstyle 4m+2}$, ${\textstyle 4m+3}$ resp., sive prout residuum minimum ipsius ${\textstyle g^{\lambda }}$ in ${\textstyle A}$, ${\textstyle B}$, ${\textstyle C}$, ${\textstyle D}$ resp. reperitur. Hinc nanciscimur criterium persimplex ad diiudicandum, ad quam classem numerus datus ${\textstyle h}$ per ${\textstyle p}$ non divisibilis referendus sit; pertinebit scilicet ${\textstyle h}$ ad ${\textstyle A}$, ${\textstyle B}$, ${\textstyle C}$ vel ${\textstyle D}$, prout potestas ${\textstyle h^{{\frac {1}{4}}(p-1)}}$ secundum modulum ${\textstyle p}$ numero ${\textstyle 1}$, ${\textstyle f}$, ${\textstyle -1}$ vel ${\textstyle -f}$ congrua evadit.

Tamquam corollarium hinc sequitur, ${\textstyle -1}$ semper ad classem ${\textstyle A}$ referri, quoties ${\textstyle p}$ sit formae ${\textstyle 8n+1}$, ad classem ${\textstyle C}$ vero, quoties ${\textstyle p}$ sit formae ${\textstyle 8n+5}$. Demonstratio huius theorematis a theoria residuorum potestatum independens ex iis, quae in Disquisitionibus Arithmeticis art. 115, III docuimus, facile adornari potest.

10.

Quum omnes radices primitivae pro modulo ${\textstyle p}$ prodeant e residuis potestatum ${\textstyle g^{\lambda }}$, accipiendo pro ${\textstyle \lambda }$ omnes numeros ad ${\textstyle p-1}$ primos, facile perspicitur, illas inter complexus ${\textstyle {B}}$ et ${\textstyle {D}}$ aequaliter dispertitas fore, basi ${\textstyle {g}}$ semper in ${\textstyle {B}}$ contenta. Quodsi loco numeri ${\textstyle g}$ radix alia primitiva e complexu ${\textstyle B}$ pro basi accipitur, classificatio eadem manebit; si vero radix primitiva e complexu ${\textstyle D}$ tamquam basis adoptatur, classes ${\textstyle B}$ et ${\textstyle D}$ inter se permutabuntur.