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Biquadratic Reciprocity Theorem

x^4\equiv q\ \left({{\rm mod\ } {p}}\right).
\end{displaymath} (1)

This was solved by Gauß using the Gaussian Integers as
\end{displaymath} (2)

where $\pi$ and $\sigma$ are distinct Gaussian Integer Primes,
\end{displaymath} (3)

and $N$ is the norm.

\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} (4)

where solvable means solvable in terms of Gaussian Integers.

See also Reciprocity Theorem

© 1996-9 Eric W. Weisstein