The following theorem on biquadratic reciprocity is proved: if
p
≡
q
≡
1
(
mod
4
)
p \equiv q \equiv 1 \pmod 4
are primes for which
(
p
|
q
)
=
1
(p|q) = 1
, and if
p
=
r
2
+
q
s
2
p = {r^2} + q{s^2}
for some integers r and s, then
\[
(
p
|
q
)
4
(
q
|
p
)
4
=
1
,
if
q
≡
1
(
mod
8
)
;
=
(
−
1
)
s
,
if
q
≡
5
(
mod
8
)
.
\begin {array}{*{20}{c}} \hfill {{{(p|q)}_4}{{(q|p)}_4} = 1,\qquad {\text {if}}\;q \equiv 1\;\pmod 8;} \\ \hfill { = {{( - 1)}^s},\quad {\text {if}}\;q \equiv 5 \pmod 8.} \\ \end {array}
\]
Simple expressions for the biquadratic character of some small primes are also obtained.