In algebraic number theory Eisenstein's reciprocity law is a reciprocity law that extends the law of quadratic reciprocity and the cubic reciprocity law to residues of higher powers.
It is one of the earliest and simplest of the higher reciprocity laws, and is a consequence of several later and stronger reciprocity laws such as the Artin reciprocity law.
It was introduced by Eisenstein (1850), though Jacobi had previously announced (without proof) a similar result for the special cases of 5th, 8th and 12th powers in 1839.
be the ring of integers of the m-th cyclotomic field
is a primitive m-th root of unity.
is called primary[2][3] if it is not a unit, is relatively prime to
, and is congruent to a rational (i.e. in
The following lemma[4][5] shows that primary numbers in
are analogous to positive integers in
are relatively prime to
which appears in the definition is most easily seen when
is totally ramified in
the m-th power residue symbol for
is either zero or an m-th root of unity: It is the m-th power version of the classical (quadratic, m = 2) Jacobi symbol (assuming
are relatively prime): Let
an integer relatively prime to
be primary (and therefore relatively prime to
is also relatively prime to
Then[8][9] The theorem is a consequence of the Stickelberger relation.
[10][11] Weil (1975) gives a historical discussion of some early reciprocity laws, including a proof of Eisenstein's law using Gauss and Jacobi sums that is based on Eisenstein's original proof.
is an arbitrary algebraic number field containing the
-th roots of unity for a prime
for pairwise relatively prime integers (i.e. in
This is the first case of Fermat's Last Theorem.
) Eisenstein reciprocity can be used to prove the following theorems (Wieferich 1909)[13][14] Under the above assumptions,
(Furtwängler 1912)[16][17] Under the above assumptions, for every prime
(Furtwängler 1912)[18] Under the above assumptions, for every prime
(Vandiver)[19] Under the above assumptions, if in addition
Eisenstein's law can be used to prove the following theorem (Trost, Ankeny, Rogers).
is solvable for all but finitely many primes