Eisenstein reciprocity


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, though Jacobi had previously announced a similar result for the special cases of 5th, 8th and 12th powers in 1839.

Background and notation

Let be an integer, and let be the ring of integers of the m-th cyclotomic field where is a primitive m-th root of unity.
The numbers are units in

Primary numbers

A number is called primary if it is not a unit, is relatively prime to, and is congruent to a rational integer
The following lemma shows that primary numbers in are analogous to positive integers in
Suppose that and that both and are relatively prime to Then
  • There is an integer making primary. This integer is unique
  • if and are primary then is primary, provided that is coprime with.
  • if and are primary then is primary.
  • is primary.

The significance of the that appears in the definition is most easily seen when is a prime. In that case Furthermore, the prime ideal of is totally ramified in
and the ideal is prime of degree 1.

''m''-th power residue symbol

For 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 Jacobi symbol :
  • If and then
  • If then is not an m-th power
  • If then may or may not be an m-th power

Statement of the theorem

Let be an odd prime and an integer relatively prime to Then

Eisenstein reciprocity

Let be primary, and assume that is also relatively prime to. Then

Proof

The theorem is a consequence of the Stickelberger relation.
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.

Generalization

In 1922 Takagi proved that if is an arbitrary algebraic number field containing the -th roots of unity for a prime, then Eisenstein's law for -th powers holds in

Applications

First case of Fermat's Last Theorem

Assume that is an odd prime, that for pairwise relatively prime integers and that
This is the first case of Fermat's Last Theorem. Eisenstein reciprocity can be used to prove the following theorems
' Under the above assumptions,
' Under the above assumptions
' Under the above assumptions, for every prime
' Under the above assumptions, for every prime
Under the above assumptions, if in addition then, and

Powers mod most primes

Eisenstein's law can be used to prove the following theorem. Suppose and that where is an odd prime. If is solvable for all but finitely many primes, then