Reciprocity Theorem
If there exists a rational integer 
such that, when
, 
, and 
are positive
integers,
 |
then 
is the 
-adic residue of
, i.e., 
is an 
-adic residue of
iff 
is
solvable for 
. Reciprocity theorems relate statements
of the form "
is an 
-adic residue of
" with reciprocal statements of the
form "
is an 
-adic residue of
."
The first case to be considered was 
(the quadratic
reciprocity theorem), of which Gauss gave the first correct proof. Gauss also
solved the case 
(cubic
reciprocity theorem) using integers of
the form 
, where 
is a root of
and 
, 
are rational integers. Gauss stated the case 
(biquadratic
reciprocity theorem) using the Gaussian integers.
Proof of 
-adic reciprocity for prime 
was given by Eisenstein in 1844-50 and by Kummer
in 1850-61. In the 1920s, Artin formulated Artin's
reciprocity theorem, a general reciprocity law for all orders.
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