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.
- 1 Background and notation
- 2 Statement of the theorem
- 3 Proof
- 4 Generalization
- 5 Applications
- 6 See also
- 7 Notes
- 8 References
Background and notation
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
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):
- 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
Let be primary (and therefore relatively prime to ), and assume that is also relatively prime to Then
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..
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
- The only primes below 6.7×1015 that satisfy this are 1093 and 3511. See Wieferich primes for details and current records.
(Mirimanoff 1911) 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) Under the above assumptions, for every prime
(Vandiver) Under the above assumptions, if in addition then and
Powers mod most primes
- Quadratic reciprocity
- Cubic reciprocity
- Quartic reciprocity
- Artin reciprocity
- Wieferich's criterion
- Mirimanoff's congruence
- 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
- 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
- Eisenstein, Gotthold (1850), "Beweis der allgemeinsten Reciprocitätsgesetze zwischen reellen und komplexen Zahlen", Verhandlungen der Königlich Preußische Akademie der Wissenschaften zu Berlin (in German): 189–198, Reprinted in Mathematische Werke, volume 2, pages 712–721
- Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition), New York: Springer Science+Business Media, ISBN 0-387-97329-X
- Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Berlin: Springer Science+Business Media, ISBN 3-540-66957-4