![\begin{displaymath}
x^4\equiv q\ \left({{\rm mod\ } {p}}\right).
\end{displaymath}](b_1506.gif) |
(1) |
This was solved by Gauß
using the Gaussian Integers as
![\begin{displaymath}
\left({\pi\over\sigma}\right)_4\left({\sigma\over\pi}\right)_4=(-1)^{[(N(\pi)-1)/4][(N(\sigma)-1)/4]},
\end{displaymath}](b_1507.gif) |
(2) |
where
and
are distinct Gaussian Integer Primes,
![\begin{displaymath}
N(a+bi)=\sqrt{a^2+b^2}
\end{displaymath}](b_1508.gif) |
(3) |
and
is the norm.
![\begin{displaymath}
\left({\alpha\over \pi}\right)_4=\cases{ 1 & if $x^4\equiv \...
...right)$\ is solvable\cr -1, i, {\rm\ or\ } -i & otherwise,\cr}
\end{displaymath}](b_1509.gif) |
(4) |
where solvable means solvable in terms of Gaussian Integers.
See also Reciprocity Theorem
© 1996-9 Eric W. Weisstein
1999-05-26