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 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

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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

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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, provided that   is coprime with  .
  • 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   

 

  and the ideal       is prime of degree 1.[6][7]

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):

  • 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

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

First supplement

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   [8]

Second supplement

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   [8]

Eisenstein reciprocity

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Let    be primary (and therefore relatively prime to    ), and assume that     is also relatively prime to   . Then[8][9]

 

Proof

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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.

Generalization

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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

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First case of Fermat's Last Theorem

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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 6.7×1015 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

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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

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Notes

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  1. ^ Lemmermeyer, p. 392.
  2. ^ Ireland & Rosen, ch. 14.2
  3. ^ Lemmermeyer, ch. 11.2, uses the term semi-primary.
  4. ^ Ireland & Rosen, lemma in ch. 14.2 (first assertion only)
  5. ^ Lemmereyer, lemma 11.6
  6. ^ Ireland & Rosen, prop 13.2.7
  7. ^ Lemmermeyer, prop. 3.1
  8. ^ a b c Lemmermeyer, thm. 11.9
  9. ^ Ireland & Rosen, ch. 14 thm. 1
  10. ^ Ireland & Rosen, ch. 14.5
  11. ^ Lemmermeyer, ch. 11.2
  12. ^ Lemmermeyer, ch. 11 notes
  13. ^ Lemmermeyer, ex. 11.33
  14. ^ Ireland & Rosen, th. 14.5
  15. ^ Lemmermeyer, ex. 11.37
  16. ^ Lemmermeyer, ex. 11.32
  17. ^ Ireland & Rosen, th. 14.6
  18. ^ Lemmermeyer, ex. 11.36
  19. ^ Ireland & Rosen, notes to ch. 14
  20. ^ 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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  • 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