Iam si potestas
secundum theorema binomiale evolvitur, per lemma praec. fiet
![{\displaystyle \Sigma (z^{4}+1)^{{\frac {1}{4}}(p-1)}\equiv -2{\pmod {p}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/206aade40798def36cbb55a26a5a3281215c960a)
Sed residua minima omnium
![{\textstyle z^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f4c61718f480fe1ddf33b2d25d07bbdbff8978f)
exhibent omnes numeros
![{\textstyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a118c6ad00742b3f5dccd2f0e74b5e369df6fd31)
, quovis quater occurrente; habebimus itaque inter residua minima ipsius
![{\displaystyle {\begin{aligned}&4(00){\text{ ad }}A\\&4(01){\text{ ad }}B\\&4(02){\text{ ad }}C\\&4(03){\text{ ad }}D\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46ed3c8b6abdb7a74381f90beca9d5537db0676b)
pertinentia, quatuorque erunt
![{\textstyle =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58f5a5ba29f6f736c98bb49181e1ab11941cc05c)
(puta pro
![{\textstyle z^{4}\equiv p-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3a6f8beff655ca86308ddc61efd4601f1eb1fe7)
). Hinc, considerando criteria complexuum
![{\textstyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a118c6ad00742b3f5dccd2f0e74b5e369df6fd31)
,
![{\textstyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de0b47ffc21636dc2df68f6c793177a268f10e9b)
,
![{\textstyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dca76d9ff4b48256b6a4a99bcb234b64b2fa72b)
,
![{\textstyle D}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e5200f518cb5afe304ec42ffdd4f6c63c702f77)
, deducimus
![{\displaystyle \Sigma (z^{4}+1)^{{\frac {1}{4}}(p-1)}\equiv 4(00)+4f(01)-4(02)-4f(03)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75830daf477e8d039421849d17cf32080df1b694)
adeoque
![{\displaystyle -2\equiv 4(00)+4f(01)-4(02)-4f(03)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/436c0f4d9efb5c3e3f00d066cf8286da1aa21590)
sive substitutis pro
![{\textstyle (00)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/679ceec6e6cc5fc3e8dd00d0e3d8f57fc7e0a1a0)
,
![{\textstyle (01)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f0d7a8e701995c84beb9d615d38ef7e37466be1)
etc. valoribus in art. praec inventis,
![{\displaystyle -2\equiv -2a-2-2bf}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24d51895bfa2733e70bf09fe69f6447f173a9487)
Hinc itaque colligimus, semper fieri debere
![{\textstyle a+bf\equiv 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a98edf69a5c6046c27bf86a0a53be56131590777)
, sive, multiplicando per
![{\textstyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b77076edca76caf3331d0551d1645b8f678283)
,
![{\displaystyle b\equiv af}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7cbd4f2193ee9e417c20f03a40706a7a132e650)
quae congruentia determinationi signi ipsius
![{\textstyle b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73a780b69dfc55238880ef18a134dc65260877e2)
, si numerus
![{\textstyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b77076edca76caf3331d0551d1645b8f678283)
iam electus est, vel determinationi numeri
![{\textstyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b77076edca76caf3331d0551d1645b8f678283)
, si signum ipsius
![{\textstyle b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73a780b69dfc55238880ef18a134dc65260877e2)
aliunde praescribitur, inservit.
20.
Postquam problema nostrum pro modulis formae
complete solvimus, progredimur ad casum alterum, ubi
est formae
: quem eo brevius absolvere licebit, quod omnia ratiocinia parum a praecedentibus differunt.
Quum pro tali modulo
ad classem
pertineat, complementa numerorum complexuum
,
,
,
ad summam
, in classibus
,
,
,
resp. contenta erunt. Hinc facile colligitur