# 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 Template:Harvs, though Jacobi had previously announced (without proof) a similar result for the special cases of 5th, 8th and 12th powers in 1839.^{[1]}

## 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 (There are other units as well.)

### Primary numbers

A number is called **primary**^{[2]}^{[3]} if it is not a unit, is relatively prime to , and is congruent to a rational (i.e. in ) integer

The following lemma^{[4]}^{[5]} 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.
- if and are primary then is primary.>
- is primary.

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

*m*-th power residue symbol

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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 (quadratic, *m* = 2) Jacobi symbol (assuming and are relatively prime):

## Statement of the theorem

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

### First supplement

### Second supplement

### Eisenstein reciprocity

Let be primary (and therefore relatively prime to ), and assume that is also relatively prime to Then

## Proof

The theorem is a consequence of the Stickelberger relation.^{[10]}^{[11]}

Template:Harvtxt 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 ^{[12]}

## Applications

### First case of Fermat's last theorem

Assume that is an odd prime, that for pairwise relatively prime integers (i.e. in ) and that

This is the first case of Fermat's last theorem. (The second case is when ) Eisenstein reciprocity can be used to prove the following theorems

**(Wieferich 1909)**^{[13]}^{[14]} Under the above assumptions,

- The only primes below 6.7×10
^{15}that satisfy this are 1093 and 3511. See Wieferich primes for details and current records.

**(Mirimanoff 1911)**^{[15]} Under the above assumptions

- Analogous results are true for all primes ≤ 113, but the proof does not use Eisenstein's law. See Wieferich prime#Connection with Fermat's last theorem.

**(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
then
and

### Powers mod most primes

Eisenstein's law can be used to prove the following theorem (Trost, Ankeny, Rogers).^{[20]} Suppose
and that
where
is an odd prime. If
is solvable for all but finitely many primes
then

## See also

- Quadratic reciprocity
- Cubic reciprocity
- Quartic reciprocity
- Artin reciprocity
- Wieferich's criterion
- Mirimanoff's congruence

## Notes

- ↑ Lemmermeyer, p. 392.
- ↑ Ireland & Rosen, ch. 14.2
- ↑ Lemmermeyer, ch. 11.2, uses the term
**semi-primary.** - ↑ Ireland & Rosen, lemma in ch. 14.2 (first assertion only)
- ↑ Lemmereyer, lemma 11.6
- ↑ Ireland & Rosen, prop 13.2.7
- ↑ Lemmermeyer, prop. 3.1
- ↑
^{8.0}^{8.1}^{8.2}Lemmermeyer, thm. 11.9 - ↑ Ireland & Rosen, ch. 14 thm. 1
- ↑ Ireland & Rosen, ch. 14.5
- ↑ Lemmermeyer, ch. 11.2
- ↑ Lemmermeyer, ch. 11 notes
- ↑ Lemmermeyer, ex. 11.33
- ↑ Ireland & Rosen, th. 14.5
- ↑ Lemmermeyer, ex. 11.37
- ↑ Lemmermeyer, ex. 11.32
- ↑ Ireland & Rosen, th. 14.6
- ↑ Lemmermeyer, ex. 11.36
- ↑ Ireland & Rosen, notes to ch. 14
- ↑ Ireland & Rosen, ch. 14.6, thm. 4. This is part of a more general theorem: Assume for all but finitely many primes Then i) if then but ii) if then or

## References

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