Pagina:Werkecarlf02gausrich.djvu/148

$${\begin{array}{c|l}{\text{character}}&{\text{modulis}}\\\hline 0&-1+6i,-1-6i,-7,-5+6i,-5-6i,-11,11+6i,11-6i\\1&-1-2i,1-4i,-5+2i,5+4i,7+2i,5-8i,-1+10i,-7-8i,\\&\quad -11-4i,7-10i\\2&3+2i,3-2i,-3+8i,-3-8i,9+4i,3+10i,3-10i\\3&-1+2i,1+4i,-5-2i,5-4i,7-2i,5+8i,-1-10i,-7+8i,\\&\quad -11+4i,7+10i\\\end{array}}$$

Revocatis his modulis ad residua minima secundum modulum ${\textstyle 3}$, videmus, eos, quibus respondet character ${\textstyle 0}$, esse partim ${\textstyle \equiv 1}$, partim ${\textstyle \equiv -1}$; eos, quorum character est ${\textstyle 1}$, fieri vel ${\textstyle \equiv 1-i}$, vel ${\textstyle \equiv -1+i}$: eos, quorum character est ${\textstyle 2}$, fieri vel ${\textstyle \equiv i}$, vel ${\textstyle \equiv -i}$; denique eos, quibus competit character ${\textstyle 3}$, esse vel ${\textstyle \equiv 1+i}$, vel ${\textstyle \equiv -1-i}$. Ex hac itaque inductione colligimus, characterem numeri ${\textstyle -3}$ pro modulo, qui est numerus primus inter associatos primarius, identicum esse cum charactere huius ipsius numeri, dum ${\textstyle 3}$, sive, quod eodem redit, ${\textstyle -3}$ tamquam modulus consideratur.

67.

Simili inductione circa alios numeros primos instituta, invenimus, numeros ${\textstyle 3\pm 2i}$, ${\textstyle -1\pm 6i}$, ${\textstyle 7\pm 2i}$, ${\textstyle -5\pm 6i}$ etc. suppeditare theoremata ei similia, ad quod in art. 65 respectu numeri ${\textstyle -1+2i}$ pervenimus; contra numeros ${\textstyle 1\pm 4i}$, ${\textstyle 5\pm 4i}$, ${\textstyle -3\pm 8i}$, ${\textstyle 5\pm 8i}$, ${\textstyle 9\pm 4i}$ etc. perinde se habere ut numerum ${\textstyle -3}$. Inductio itaque perducit ad elegantissimum theorema, quod ad instar theoriae residuorum quadraticorum in arithmetica numerorum realium Theorema fundamentale theoriae residuorum biquadraticorum nuncupare liceat, scilicet:

Denotantibus ${\textstyle a+bi}$, ${\textstyle a^{\prime }+b^{\prime }i}$ numeros primos diversos inter associatos suos primarios, i.e. secundum modulum ${\textstyle 2+2i}$ unitati congruos, character biquadraticus numeri ${\textstyle a+bi}$ respectu moduli ${\textstyle a^{\prime }+b^{\prime }i}$ identicus erit cum charactere numeri ${\textstyle a^{\prime }+b^{\prime }i}$ respectu moduli ${\textstyle a+bi}$, si vel uterque numerorum ${\textstyle a+bi}$, ${\textstyle a^{\prime }+b^{\prime }i}$, vel alteruter saltem, ad primum genus refertur, i.e. secundum modulum 4 unitati congruus est: contra characteres illi duabus unitatibus inter se different, si neuter numerorum ${\textstyle a+bi}$, ${\textstyle a^{\prime }+b^{\prime }i}$ ad primum genus refertur, i.e. si uterque secundum modulum 4 congruus est numero ${\textstyle 3+2i}$.